Relation between nonorientability of the Möbius strip and the Möbius bundle

Question

There are two ways in which the open Möbius strip $M$ is related to orientability:

1. $M$ is nonorientable as a manifold;
2. $M$ is the total space of the nonorientable line bundle $M \to S^1$.

Is there any relation between these two concepts? If so, can it be generalized to arbitrary vector bundles?

Is a statement like "A noncompact manifold $M$ is nonorientable iff it is the total space of a nonorientable vector bundle" unreasonable?