It is true that if a Manifold $M$ with boundary is orientable then its boundary $\partial M$ is also orientable. I'm wondering if the converse is also true. Of fact, there are numerous minor examples where the outcome does not hold. When the manifold is not connected, for example, one may select the disjoint union of a non-orientable manifold without boundary and an orientable manifold with boundary. Its boundary is orientable, but it has a non-orientable submanifold and thus is not orientable.
I'm curious whether the converse holds true in more complex circumstances, or if there are any intriguing counterexamples.