Let $M$ be a manifold with boundary. Hatcher writes that a compact manifold with boundary is $R$-orientable if $M - \partial M$ is $R$-orientable. That is there exists a function $x \to \mu_x \in H_n(M \vert x)$ that satisfies the local consistency property.
However, I have seen other sources talk of the orientation of the boundary induced by the orientation on the interior, i.e., the 'induced orientation'.
How do we extend the function $x \to \mu_x$ onto the boundary so to define a orientation on the boundary of a manifold?