Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing automorphisms (e.g. $\mathbb{C}\mathbb{P}^n$ for even $n$), I'm looking for an answer to the following question:
What topological conditions ensure that $M$ admits an orientation-reversing automorphism (homeomorphism, diffeomorphism,...)?