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It was recently demonstrated that monolayer FeSe on a SrTiO₃ substrate is a superconductor with Tc between 60 and 100 K, compared to 8 K in bulk FeSe. This is in contrast to the expected behavior; thinning a superconductor typically reduces T_c. Similar results have been obtained in monolayer FeSe deposited on BaTiO₃ and on anatase or rutile TiO₂. Here we determine the atomic structure of an interfacial layer and identify a possible 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 long-range magnetic and orbital order not found in the typical TiO₂ surface termination nor in previously identified surface reconstructions.