On a Riemannian manifold, we can see concrete realizations of the real cohomology groups as spaces of harmonic forms. On a closed manifold, the concrete realizations of the pth cohomology and the (n-p)th cohomology inside the space of p-forms coincide. This is not true on manifolds with boundary; both the pth and (n-p)th cohomology can still be realized as subspaces of the space of p-forms, but these subspaces only meet at the origin. In many interesting cases, the principal angles between these subspaces are acute, yielding invariants of Riemannian manifolds with boundary.