Your question is one about true mathematical duality, you just do not know it. What you are looking for is Hodge duality, which holds in the exterior algebra of any vector space, and the differential forms one looks at in EM, GR and elsewhere are just elements of the exterior algebra of the tangent space (or, equivalently, of deRham cohomology).

It is a basic result (its combinatorial - if you can count, you can prove it!) that the $p$-th degree of the exterior algebra over a space of dimension $n$ has dimension $n \choose p$, and since $\binom{n}{p} = \binom{n}{n-p}$, the spaces of the $p$-th and $n-p$-th degree have indeed the same dimension, and sending their basis vectors to each other via the epsilon tensor defines an isomorphism.

This post imported from StackExchange Physics at 2014-06-29 09:35 (UCT), posted by SE-user ACuriousMind