Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the resulting space is again an oriented manifold, I think I need to take an orientation reversing homeomorphism between boundaries.
So let $f:\partial M_1 \to \partial M_2$ be a homeomorphism.
My question is that:should $f$ be an orientation preserving with respect to the induced orientation on the boundaries? Or can we choose any orientations on the boundaries so that $f$ is orientation reversing with respect to the choice of the orientations of the boundaries?