Reciprocity/duality theorems may give you unexpected results if you don't expect the connections.
Dan Ranmas already mentioned Poincare duality. To clarify, Poincare duality is not just abstract nonsense. It fails for non-manifolds like general abstract simlicial complexes. For a [mod $2$] oriented manifold of dimension $d$, the [mod $2$] homology in dimension $k$ is isomorphic to the [mod $2$] homology in dimension $d-n$.d-k$.
Quadratic reciprocity relates whether $p$ is a square mod $q$ with whether $q$ is a square mod $p$.
Weil reciprocity relates the values of a rational function $f$ at the zeros and poles of $g$ with the values of $g$ at the zeros and poles of $f$.
Stanley reciprocity relates a generating function for the lattice points in a convex cone with a generating function for the lattice points in the interior evaluated at reciprocal arguments.
