As a result of their applications in network coding, space-time coding, and coding for criss-cross errors, matrix codes have garnered significant attention; in various contexts, these codes have also been termed rank-metric codes, space-time codes over finite fields, and array codes. We focus on characterizing matrix codes that are both efficient (have high rate) and effective at error correction (have high minimum rank-distance). It is well known that the inherent trade-off between dimension and minimum distance for a matrix code is reversed for its dual code; specifically, if a matrix code has high dimension and low minimum distance, then its dual code will have low dimension and high minimum distance. With an aim towards finding codes with a perfectly balanced trade-off, we study self-dual matrix codes. In this work, we develop a framework based on double cosets of the matrix-equivalence maps to provide a complete classification of the equivalence classes of self-dual matrix codes, and we employ this method to enumerate the equivalence classes of these codes for small parameters.