I'd count among my favourites:

* the Schroeder-Bernstein theorem in set theory

* the fact that the axiom of choice is equivalent to the statement that every set is well-ordered: how can something so intuitively correct and something so intuitively wrong be equivalent?

* the general notion of dualism in category theory. Two for the price of one!

* the ways fixpoints can be used in TCS: the Y combinator (heh), and the Kleene and Knaster-Tarski theorems. The best part is that they're all intuitively the same.