We describe measurements and analysis of the electronic spin relaxation in a short-period InAs/GaSb superlattice. Intrinsic inversion symmetry breaking due to tetrahedral bonding at the superlattice interfaces leads to spin relaxation over an order of magnitude greater than that expected from a continuum model. We extract the inversion asymmetry of the bonding character at the two distinct interfaces in this superlattice and describe how these (and future) measurements can be used to constrain fundamental interface parameters of superlattice envelope-function models.