Are there odd-sheeted coverings of non-orientable surfaces by orientable surfaces? For any non-orientable surface (compact,connected) $X$ with genus $h$, we have a $2n$-sheeted cover of $X$ by an orientable surface $Y$ first by covering $X$ by $\Sigma_{h-1}$ (a double cover) and then taking an $n$-sheeted orientable cover of $\Sigma_{h-1}$ and composing. Also there is an $n$-sheeted cover of $X$ by a non-orientable manifold, and a double cover of that by an orientable manifold. These give isomorphic coverings. My question is that do all coverings of non-orientable surfaces by orientable surfaces arise in this fashion? Specifically, I want to know if there are odd-sheeted coverings of non-orientable surfaces by orientable surfaces, and if not why there are none. (It seems highly unlikely that there are any to me, and it seems a clever algebraic trick should show this, but I do not know how to do it.) If this can be done without cohomology, I would appreciate it.

I don't think Daniel's link is really a propos -- Mednykh is solving a much harder problem. It is fairly easy to show that [every covering map from an orientable surface [this holds in much greater generality] to an orientable surface factors through the orientatation cover.]( Since the orientation cover is a double cover you are done.