Every manifold has an orientable double cover; it disconnects exactly when the base is orientable.Suggest a slogan
Every \(n\)-manifold \(M\) has an orientable two-sheeted covering space \(\tilde M\to M\), the orientation double cover [@Hat02]. If \(M\) is connected, then \(M\) is orientable if and only if \(\tilde M\) has two components; in particular, \(M\) is orientable if \(\pi_1(M)\) has no subgroup of index two [@Hat02].