*

It was recently demonstrated that monolayer FeSe on SrTiO₃ is a superconductor with T_c between 60 and 100 K, compared to 8 K in bulk FeSe. In contrast, T_c has been measured to be only 3.7 K in bi-layer FeSe deposited on graphene (extrapolated to about 2 K in a monolayer), pointing to the major role of the interface in enhancing superconductivity. Here we determine the atomic structure of an interfacial layer and identify its role in driving the increase in T_c using a combination of quantum mechanical calculations and scanning transmission electron microscopy. Within our DFT calculations, this interfacial layer hosts magnetic and orbital order not found in the typical TiO₂ surface termination nor in other surface reconstructions. Interactions between this interfacial monolayer and FeSe generate symmetry-breaking distortions in the film that are favorable for increasing T_c and are not present in other possible FeSe / STO interface structures. We propose that this may provide a path forward toward the design and enhancement of other two-dimensional superconductors.