The neutrally-charged silicon vacancy in diamond is a promising system for quantum technologies that combines high-efficiency, broadband optical spin polarization with long spin lifetimes (T2 ~ 1 ms at 4 K) and up to 90% of optical emission into its 946 nm zero-phonon line. However, the electronic structure of SiV0 is poorly understood, making further exploitation difficult. Performing photoluminescence spectroscopy of SiV0 under uniaxial stress, we find the previous excited electronic structure of a single 3A1u state is incorrect, and identify instead a coupled 3Eu - 3A2u system, the lower state of which has forbidden optical emission at zero stress and so efficiently decreases the total emission of the defect: we propose a solution employing finite strain to form the basis of a spin-photon interface. Isotopic enrichment definitively assigns the 976 nm transition associated with the defect to a local mode of the silicon atom.