For the question about homeomorphisms the answer is no, even if $X$ and $X'$ are algebraic surfaces.
In fact, in his paper [Orientation reversing homeomorphisms in surface geography, Math. Ann. 292 (1992)], D. Kotschick proves the following result:
Theorem. There exist infinitely many pairs of simply connected algebraic surfaces of general type which are orientation-reversing homeomorphic (with respect to their complex orientations), but not diffeomorphic.
He also makes a conjecture about orientation-reversing diffeomorphic algebraic surfaces. As I said in my comments before, by using Seiberg-Witten theory one proves that, given any diffeomorphism $\phi \colon X \to X'$ between two smooth $4$-manifolds, one has either $\phi(K_X)=K_{X'}$ or $\phi(K_X)=-K_{X'}$.
Kotschick's conjecture is therefore the following:
Conjecture. If two algebraic surface with finite fundamental group are orientation-reversing diffeomorphic, then they are homeomorphic to a geometrically ruled rational surface. In particular, they are simply connected.
I do not know the current state of this conjecture.
Added On February 29, 2012. D. Kotschick kindly informed me that he actually proved this conjecture in his paper Orientetions and geometrizations of compact complex surfaces, Bulletin of the London Mathematical Society 29 (1997), 145-149.