The interface of insulating oxides SrTiO₃ and LaAlO₃ exhibits a 2D electron gas featuring intriguing transport phenomena, including superconductivity, controllable by gate voltage. Our experiments surprisingly show that while the superconducting transition temperature and inverse Hall coefficient are non-monotonic as function of gate in both the (100) and (111) versions of the interface, the frequency of the Shubnikov-de Haas (SdH) oscillations is non-monotonic only in the former system. We developed a theoretical model, incorporating band structure with the appropriate crystal symmetries and electronic correlations, which allows for competition between the bands. Using it we show that bands with different orbital character arise at the two interfaces, and how their competition results in non-monotonic population of the upper band in (100) but not in (111), in accordance with the SdH data. The resulting Fermi surface structure can explain the other experimental measurements.