Quantum confinement at complex oxide interfaces establishes an intricate hierarchy of the strongly correlated d orbitals which is widely recognized as a source of emergent physics. The most prominent example is the (001) LaAlO3/SrTiO3 (LAO/STO) interface, which features a dome-shaped phase diagram of superconducting critical temperature and spin-orbit coupling (SOC) as a function of electrostatic doping, arising from a selective occupancy of t2g orbitals of different character. Here we study (111)-oriented LAO/STO interfaces, where the three t2g orbitals contribute equally to the subband states caused by confinement, and investigate the impact of this unique feature on electronic transport. We show that transport occurs through two sets of electronlike subbands, and the carrier density of one of the sets shows a nonmonotonic dependence on the sample conductance. Using tight-binding modeling, we demonstrate that this behavior stems from a band inversion driven by on-site Coulomb interactions. The balanced contribution of all t2g orbitals to electronic transport is shown to result in strong SOC with reduced electrostatic modulation.