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][1],  Bulletin of the London Mathematical Society **29** (1997), 145-149. 


  [1]: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=18825