If $M$ is a compact, oriented, $n$ dimensional manifold, then there is a canonical (though non-natural) isomorphism $$D:H^q(M,\Z)\to H_{n-q}(M,\Z)$$ (where $H^k(M,\Z)$ is the $k$ homology group of $M$ with integer coefficients and $H_k(M,\Z)$ the $k$ cohomology group) for all $q$ which is given by cap product with a generator of $H_n(M,\Z)$ (a choice of a generator here corresponds to an orientation). This isomorphism exists with coefficients in $\mathbb{Z}/2\mathbb{Z}$ regardless of orientation.

This isomorphism gives a nice interpretation to cup product. If $X,Y$ are transverse submanifolds of $M$ then $X\cap Y$ is also a submanifold. All of these submanifolds represent homology classes of $M$ in the appropriate dimensions, and $$D^{-1}([X])\cup D^{-1}([Y])=D^{-1}([X\cap Y]),$$ where $\cup$ is cup product, and $\cap$ in intersection, not cap product.