Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore homology of $X$, which is isomorphic to the cohomology of $X$. What is the simplest definition of $[Y]$ in the general case (where $X$ and $Y$ are not necessarily oriented)?
Note that a simple generalization of this question would be to ask how to define the pushforward in cohomology along a proper oriented map. (Then $[Y]$ would simply be the pushforward of $1\in H^0(Y)$.) I would be happy to know the answer to this more general question, but I asked the simpler version to be as concrete as possible.