In the History of Middle-Earth book collection Myths Transformed we learn a lot about how Sauron perceived the Istari and what he knew of them.
Sauron had, in fact been very like Saruman, and so understood him quickly and could guess what he would be likely to think and do, even without the aid of the palantiri or of spies; whereas Gandalf eluded and puzzled him.
(Home X: Myths Transformed)
Gandalf he did not understand. But certainly he had already become evil, and therefore stupid, enough to imagine that his different behaviour was due simply to weaker intelligence and lack of firm masterful purpose. He was only rather a cleverer Radagast — cleverer, because it is more profitable (more productive of power) to become absorbed in the study of people than of animals.
My question now is how does Gandalf elude and puzzle Sauron so much? We know he found Saruman simple to guess down to having similar minds (maybe down to both being maia of Aule?), we also know he must know a fair bit of Radagast as he refers to Gandalf as merely a smarter version of Radagast who was known to have abandoned his task which the Istari were given.
While Sauron went face to face with Saruman via the palantir, it is surprising that he finds Gandalf basically strange, even though Gandalf in reality does the most to circumvent Sauron out of the Istari in the north-west.
- Gandalf enters Dol Guldur which Sauron flees.
- Gandalf enters Dol Guldur again in secret and claims he only just escaped with his life, which suggests Sauron may have ended up noticing him.
- Gandalf pushes White Council to attack Dol Guldur which Sauron foresees.
- Gandalf fought with all nine of Sauron's greatest servants at Weathertop and played a part in destroying their physical bodies temporarily.
- Gandalf leads the Fellowship from Rivendell, which Sauron also knows.
- Gandalf strives with the Dark Tower when Frodo puts the ring on at Amon Hen.
- Gandalf is present at the Battle of the Black Gate, in which the mouth of Sauron declares they know all about Gandalf.
As I've pointed out, Gandalf is clearly sent to contest Sauron. But I don't understand why Sauron does not see this; throughout the Third Age, well, since the coming of the Istari Gandalf has played a role in almost every contest against Sauron.
So how did he elude and puzzle him?