An emerging class of semiconductor heterostructures involves stacking discrete monolayers such as the transition metal dichalcogenides (TMDs) to form van der Waals heterostructures. In these structures, it is possible to create interlayer excitons (ILEs), spatially indirect, bound electron-hole pairs with the electron in one TMD layer and the hole in an adjacent layer. We are able to clearly resolve two distinct emission peaks separated by 24 meV from an ILE in a MoSe2/WSe2 heterostructure fabricated using state-of-the-art preparation techniques. These peaks have nearly equal intensity, indicating they are of common character, and have opposite circular polarizations when excited with circularly polarized light. Ab initio calculations successfully account for these observations - they show that both emission features originate from excitonic transitions that are indirect in momentum space, are split by spin-orbit coupling, and that including interlayer hybridization is essential in correctly describing the ILE transition. Although well separated in momentum space, we find that in real space the electron has significant weight in both the MoSe2 and WSe2 layers, contrary to the commonly assumed model. This is a significant consideration for understanding the static and dynamic properties of TMD heterostructures.