In the 'Voice of Saruman' chapter, Gandalf the White tells Saruman that he may leave Orthanc to go wherever he chooses,
even to Mordor, if Saruman so wishes. What was Gandalf thinking?

First, as a caveat, I'm not sure how well received Saruman would be at Barad-dur - if I were Sauron, I'd definitely watch my back. That said, Saruman had valuable information that could have aided Sauron in his war effort and in his efforts to recover the Ring.
So how would that play out? And what would the West face if Saruman did physically join up with the East, if Hand and Eye were to join forces.