  The interface between the two insulating oxides SrTiO$_3$ and LaAlO$_3$ gives
rise to a two-dimensional electron system with intriguing transport phenomena,
including superconductivity, which are controllable by a gate. Previous
measurements on the (001) interface have shown that the superconducting
critical temperature, the Hall density, and the frequency of quantum
oscillations, vary nonmonotonically and in a correlated fashion with the gate
voltage. In this paper we experimentally demonstrate that the (111) interface
features a qualitatively distinct behavior, in which the frequency of
Shubnikov-de Haas oscillations changes monotonically, while the variation of
other properties is nonmonotonic albeit uncorrelated. We develop a theoretical
model, incorporating the different symmetries of these interfaces as well as
electronic-correlation-induced band competition. We show that the latter
dominates at (001), leading to similar nonmonotonicity in all observables,
while the former is more important at (111), giving rise to highly curved Fermi
contours, and accounting for all its anomalous transport measurements.
