Among the best-known examples of nonorientable, compact manifolds are projective spaces. However for these one has the fact that $\mathbb RP^n$ is orientable iff $n$ is odd, so that only "half" of them are nonorientable. This statement relies on the fact that the antipodal map $\alpha$ on $S^n$ is orientation-preserving iff $n$ is odd, and $\mathbb RP^n = S^n/\{id_{S^n}, \alpha\}$ is a quotient manifold.
Now, since $S^n$ is precisely the orientation covering of $\mathbb R P^n$, and $\alpha$ is the nontrivial covering transformation on $S^n$, I wondered, whether one could generalise this to arbitrary compact, connected manifolds $M^n$ of dimension $n$. I.e. whether it is true that if $n$ is odd, the nontrivial covering transformation cannot be orientation reversing, which would mean that every such $M$ has to be orientable.
I guess this cannot be true so generally, so I went looking for some counterexamples, but neither I myself nor google seems to find any. Would anyone here be able to help me out with an example, maybe?
Thanks in advance,
Sam