  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)
LaAlO$_3$/SrTiO$_3$(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
$t_{2g}$ orbitals of different character. Here we study (111)-oriented LAO/STO
interfaces - where the three $t_{2g}$ orbitals contribute equally to the
sub-band states caused by confinement - and investigate the impact of this
unique feature on electronic transport. We show that transport occurs through
two sets of electron-like sub-bands, and the carrier density of one of the sets
shows a non-monotonic 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 $t_{2g}$
orbitals to electronic transport is shown to result in strong SOC with reduced
electrostatic modulation.
