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.