Does Saruman invade Rohan to please Sauron, or is it a desperate last-ditch attempt to recover the ring?

Question

I don't really understand most of the politics of Sauron and Saruman during the time-period between the departure of the fellowship from Lórien and the moment where Aragorn shows himself in Orthanc's palantír to Sauron.

For instance, why does Saruman decide to have a large host of orcs and wild-men invade Rohan (with a blow against Helm's Deep in particular), shortly after he finds out about the destruction of his expedition force carrying Merry and Pippin?

There's a relevant paragraph about this when Gandalf rejoins with Aragorn, Legolas & Gimli.

'Then is not Saruman a traitor?' said Gimli.

'Indeed yes,' said Gandalf. 'Doubly. And is not that strange? Nothing that we have endured of late has seemed so grievous as the treason of Isengard. Even reckoned as a lord and captain Saruman has grown very strong. He threatens the Men of Rohan and draws off their help from Minas Tirith, even as the main blow is approaching from the East. Yet a treacherous weapon is ever a danger to the hand. Saruman also had a mind to capture the Ring, for himself, or at least to snare some hobbits for his evil purposes. So between them our enemies have contrived only to bring Merry and Pippin with marvellous speed, and in the nick of time, to Fangorn, where otherwise they would never have come at all!

'Also they have filled themselves with new doubts that disturb their plans. No tidings of the battle will come to Mordor, thanks to the horsemen of Rohan; but the Dark Lord knows that two hobbits were taken in the Emyn Muil and borne away towards Isengard against the will of his own servants. He now has Isengard to fear as well as Minas Tirith. If Minas Tirith falls, it will go ill with Saruman.'

'It is a pity that our friends lie in between,' said Gimli. 'If no land divided Isengard and Mordor, then they could fight while we watched and waited.'

‘The victor would emerge stronger than either, and free from doubt,’ said Gandalf. 'But Isengard cannot fight Mordor, unless Saruman first obtains the Ring. That he will never do now. He does not yet know his peril. There is much that he does not know. He was so eager to lay his hands on his prey that he could not wait at home, and he came forth to meet and to spy on his messengers. But he came too late, for once, and the battle was over and beyond his help before he reached these parts. He did not remain here long. I look into his mind and I see his doubt. He has no woodcraft. He believes that the horsemen slew and burned all upon the field of battle; but he does not know whether the Orcs were bringing any prisoners or not. And he does not know of the quarrel between his servants and the Orcs of Mordor [i.e. Grishnákh's band]; nor does he know of the Winged Messenger."

So by this point in time, Saruman's politics is clearly devised to get the ring in his possession as soon as possible and he is willing to risk Sauron noticing his betrayal of their alliance for that purpose. That renders the "Saruman invaded Rohan on Sauron's orders" theory somewhat problematic. Maybe in the mind of Saruman obeying the orders of the Dark Tower now may prevent the latter from ever noticing his betrayal? If Sauron is aware of his betrayal -as we, the reader, know he is- he may be more forgiving later if Saruman now resumes the alliance in a more loyal way?

Also Sauron's calculations are difficult to understand in this theory: for military purposes (distract Rohan from helping Gondor/Minas Tirith) Isengard's invasion comes very early. In fact, we learn later that Sauron sped up his attack on Minas Tirith because of Aragorn's revelation in the Palantír (which is only after the battle of Helm's Deep). In addition, Sauron does not know yet that Merry & Pippin do not carry the Ring nor that Gandalf has returned. So he has to take into account a scenario where the Ring is in the Rohan area. In that case, his immediate worst-case scenario is: the Ring getting in Saruman's hands (Saruman being the most powerful individual in that area in terms of possibly being able to master the Ring) and Saruman claiming the Ring and mounting a challenge against Mordor. So his best strategy seems to order Saruman to refrain from aggression against Rohan in order to minimize the latter's chances to obtain the Ring.

One could also speculate that Saruman is in fact calculating that his feigned 'alliance' with Sauron offers no more hopes for him and he is desperately continuing his policy of obtaining the Ring for himself. By this time he has to resort to wild guesses about the Ring's whereabouts and, as his expedition force was destroyed by horsemen and he found no hobbit-bodies on the battle-field, he draws the following conclusions:

1) If my orc-band didn't carry any prisoners, let alone the Ring, my fate is sealed anyway. So let me proceed by betting they did carry some loot.

2) It were horsemen who destroyed the orc-band, so any prisoners (or other loot) must now be on their way to Rohan. Searching for the Ring in Rohan (via Wormtongue's manipulation of the king or by force) is therefore my best hope to secure victory after all.

Answer 1

Short answer

Saruman's goose was cooked at this point and he knew it. His invasion was a last ditch attempt to seize the ring for himself.

Detailed answer

It's hard to give a definitive answer because Tolkien rarely gives us a direct insight into the thinking of the enemy. Just about everything we know is something the hobbits saw, that they could have learned from Gandalf, Elrond, etc., or from libraries in Minas Tirith or Rivendell. However, we do know that Sauron did not trust Saruman; otherwise why send Grishnakh across the Anduin? Grishnakh confirms this himself:

Who does he [Saruman] think he is, setting up on his own with his filthy white badges? They might agree with me, with Grishnakh their trusted messenger; and I Grishnakh say this: Saruman is a fool, and a dirty treacherous fool. But the Great Eye is on him.

(The Uruk-hai)

We also know that Saruman initially intended to seize the ring for himself, for example

2851: The White Council meets. Gandalf urges an attack on Dol Guldur. Saruman overrules him. [It afterwards became clear that Saruman had then begun to desire to possess the One Ring himself, and he hoped that it might reveal itself, seeking its master, if Sauron were let be for a time.] Saruman begins to search near the Gladden fields.

(The Tale of Years)

However

c3000: Saruman dares to use the Palantir of Orthanc, but becomes ensnared by Sauron, who has the Ithil Stone. He becomes a traitor to the Council. His spies report that the Shire is being closely guarded by the Rangers.

The nature of this 'ensnarement' seems to be limited to passing information to Mordor, since Saruman later makes another attempt to seize the Ring for himself, sending Uglúk's band hundreds of miles across hostile territory to intercept the fellowship and giving explicit order that the hobbits were to be brought to Isengard. Moreover, during his captivity in Orthanc in TA3018, Gandalf notes that Saruman's forces were 'in rivalry of Sauron and not in his service yet'. Even Saruman's passing information to Sauron seems to be unreliable, as confirmed by Sauron's words via the Palantíri:

So you have come back? Why have you neglected to report for so long?

(The Palantir)

There is some additional relevant material in 'The Hunt for the Ring' in the Unfinished Tales. There are several conflicting versions of this material, but each has the black riders visiting Orthanc in TA3018 asking for information about the Ring and the Shire, and in each version Saruman lies and the Black Riders later find out. One version states that, upon the Black Riders' arrival at Isengard

His [Saruman's] dread was great, for his hope of deceiving Sauron, or at least of receiving his favour in victory, was utterly lost. Now either he himself must gain the Ring or come to ruin and torment.

It is curious that the Tale of Years makes no mention of the Nazgûls' visit to Isengard, and it may be that Tolkien abandoned this idea, so that it is not canon. Nevertheless, the overwhelming likelihood is that Saruman knew Sauron didn't trust him. Assembling his own forces and sending Uglúk's band to the Anduin were unlikely to go unnoticed by Sauron, and Saruman must have known this.

EDIT

Saruman's conversation with Gandalf in Isengard took place before the Nazgul visited. His knowledge that the latter were searching for the Shire must have come from elsewhere.

Answer 2

You ask about the strategies of both Saruman and Sauron

Saruman

I think your supposition is correct.

He believed his assault on Rohan would please Sauron. Don't forget that the open assault began on 25 February with the first Battle of the Fords of Isen. That was a day before Merry and Pippin were abducted, so Saruman would not have thought at the time that Sauron had any reason to suspect him.

25 (Feb) The Company pass the Argonath and camp at Parth Galen. First Battle of the Fords of Isen; Théodred son of Théoden slain.

26 (Feb) Breaking of the Fellowship. Death of Boromir; his horn is heard in Minas Tirith. Meriadoc and Peregrin captured. Frodo and Samwise enter the eastern Emyn Muil. Aragorn sets out in pursuit of the Orcs at evening. Éomer hears of the descent of the Orc-band from Emyn Muil.

_{The Lord of the Rings Appendix B, Section 2: The Third Age
Page 1092 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)}

As the passage you quoted points out, Saruman had no way of knowing if the Ring was present when the orcs were killed by Éomer and his men. So he would think it possible that Éomer took the Ring, and his best chance of retrieving it was to attack Rohan.

Sauron

As I mentioned, the assault on Rohan started before the abduction of Merry and Pippin. The passage quoted in the question tells us that Sauron knew that two hobbits had been taken towards Isengard (thanks @Thibaut Demaerel for reminding me). He might suspect that Saruman had the ring (or information about it) but he had no reason to think that the Rohirrim had it, so he would have no reason to tell Saruman to hold off on the attack.

It is not clear if or when Sauron learned what happened. After all, none of the orcs escaped from Éomer's attack. It is possible that Sauron's first suspicion that the Ring might be in Rohan was when Pippin looked in the palantír. Sauron believes that Pippin is with Saruman, and he tells Pippin to tell Saruman "this dainty" is not for him. Pippin describes his conversation with Sauron to Gandalf and the others.

‘“ So you have come back? Why have you neglected to report for so long?”

‘I did not answer. He said: “Who are you?” I still did not answer, but it hurt me horribly; and he pressed me, so I said: “A hobbit.”

‘Then suddenly he seemed to see me, and he laughed at me. It was cruel. It was like being stabbed with knives. I struggled. But he said: “Wait a moment! We shall meet again soon. Tell Saruman that this dainty is not for him. I will send for it at once. Do you understand? Say just that!”

_{The Lord of the Rings Book Three, Chapter 11: The Palantír
Page 593 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)}

The next communication with Saruman was via the Nazgûl (presumably on its way to collect "the dainty") that Gandalf sees two days after the Battle of Helm's Deep before he sets off with Pippin to Minas Tirith.

5 (Mar) Théoden reaches Isengard at noon. Parley with Saruman in Orthanc. Winged Nazgûl passes over the camp at Dol Baran. Gandalf sets out with Peregrin for Minas Tirith. Frodo hides in sight of the Morannon, and leaves at dusk.

_{The Lord of the Rings Appendix B, Section 2: The Third Age
Page 1093 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)}

Answer 3

Let me present evidence supporting the viewpoint that -at this point in time- both Saruman and Sauron were giving priority to the capture of the ring and would risk their alliance, if necessary, for that purpose. In that light, Saruman's speeding up of his war efforts against Rohan may have been more motivated by a potential last-ditch recovery of the Ring rather than the pleasing of Sauron (although he could later still feign his war to be in support of Sauron, if Sauron would be victorious against Gondor. Also thanks @Blackwood to point out that the battle of Isen took place one day before the abduction of Merry and Pippin)

1) Both Sauron and Saruman send sizeable expedition forces to the Anduin. (In the book, we even get to see a winged Nazgûl tracking the fellowship while they are still on the Anduin). During the chapter "the Uruk-Hai" we learn that these expedition forces have conflicting orders with regards to the destination of "the loot/prisoners". The ambush on the fellowship is not a concerted effort of the Two Towers.

2) One may perhaps speculate that Grishnákh's return to Uglúk's Isengardeners with a band of Mordor Orks is to orchestrate an ambush or coup on the Isengardeners when those let their guard down. Consider the following excerpts from the "Uruk-Hai"-chapter of the the Two Towers:

The hobbits were left with the Isengarders: a grim dark band, four score at least of large, swart, slant-eyed Orcs with great bows and short broad-bladed swords. [...] At that moment Pippin saw why some of the troop had been pointing eastward. From that direction there now came hoarse cries, and there was Grishnákh again, and at his back a couple of score of others like him: long-armed crook-legged Orcs. [...] Uglúk: 'The Whiteskins are coming. What's happened to your precious Nazgûl? Has he had another mount shot under him? Now, if you'd brought him along, that might have been useful-if these Nazgûl are all they make out.' 'Nazgûl, Nazgûl,' said Grishnákh, shivering and licking his lips, as if the word had a foul taste that he savoured painfully. 'You speak of what is deep beyond the reach of your muddy dreams, Uglúk,' he said. 'Nazgûl! Ah! All that they make out! One day you'll wish that you had not said that. Ape!' he snarled fiercely. 'You ought to know that they're the apple of the Great Eye. But the winged Nazgûl: not yet, not yet. He won't let them show themselves across the Great River yet, not too soon. They're for the War-and other purposes.'

So Grishnákh has returned with sufficiently many orcs to engage in 1 to 1-2 combat (there's perhaps a bit more Isengardeners than Mordor orcs). Add to that that Grishnákh may have been lying about the Nazgûl: perhaps he's in close pursuit and will join the fray as soon as the combined Mordor forces outmatch the Isengardeners. If this hypothesis is true, the Mordor plan is eventually thwarted by the early intervention of the Rohirrim (Remember that the Nazgûl's fell beast was shot by Legolas, so maybe his pursuit is slower than it otherwise would have been) and later by the presence of Saruman, Gandalf and the "Three Hunters" in that area. The thwarting of this coup allows Sauron to still feign alliance to Saruman later when Pippin looks in the Palantír (he's unaware that by that time Saruman was overthrown). In this scenario, Sauron does know about the Rohirrim ambush early on (e.g. the Nazgûl inspects the battlefield shortly after Gandalf and his company leave to Edoras) but he can extract very little information from the battlefield-leftovers.

3) After the downfall of Saruman, there's this famous event where Pippin looks into the Palantír recovered from the Orthanc. In it, Sauron first seems to think it is Saruman gazing into the Palantír. He says: “So you have come back? Why have you neglected to report for so long?”. Now recall that the ambush at Amon Hen was only the 26'th of February while Pippin looks in the Palantír on the 5th of March, so barely a week has gone by. If we surmise that Saruman failed to communicate with Sauron this whole week, we must conclude that the rapid escalation of Isengard's invasion during that week is all part of Saruman's personal agenda (i.e a final desperate attempt to recover the ring from Rohan. The reason why he does not want use the Palantír during this time is presumably that his mind is now so bent on his own will, yet he is full of doubts and fear. So he fears that his mind will be easily exposed to Sauron if he gazes into the Palantír in such a condition). Saruman is playing solo during this week of his downfall.

Answer 4

Saruman made an alliance with Sauron. I think he did not accepted the leadership of Sauron. He did not attack Helm's Deep to please Sauron. Once Tree Beard said to Merry and Pipin that Saruman wants to be a super power And Gandalf said to Aragon that if Mordor and Isengard fights Isengard cannot win unless Saruman holds the ring. Now Saruman fears by thinking what would happen if Theoden captures the ring

In my view the Battle of Hornburg is a effort of Saruman to prevent Theoden in capturing the ring and to make the ring himself to rule the Middle-earth.