Two Orientations

All trivializations of an Euclidean line bundle $\pi\colon\mathscr R\to B$ over a connected base $B$ split in two classes which can be naturally named orientations of $\pi$. In the case of an orienting sheaf of a manifold or a vector bundle, they admit a natural interpretation as orientations of these objects. This approach establishes an extension of standard classical constructions to all manifolds and vector bundles independently of orientability restrictions in the usual sense.