  Low density two-dimensional electron gases (2DEGs) with spin-orbit coupling
are highly sensitive to an in-plane magnetic field, which impacts their Fermi
surfaces and transport properties. Such 2DEGs, formed at transition metal oxide
surfaces or interfaces, can also undergo surface phase transitions leading to
polar metals which exhibit electronic nematicity. Motivated by experiments on
such systems, we theoretically study magnetotransport in $t_{2g}$ orbital
systems, using Hamiltonians which include atomic spin-orbit coupling (SOC) and
broken inversion symmetry, for both square symmetry (001) and hexagonal
symmetry (111) 2DEGs. Using a numerical solution to the full multiband
matrix-Boltzmann equation, together with insights gleaned from the impurity
scattering overlap matrix, we explore the anisotropic magnetoresistance (AMR)
in the presence of impurities which favor small momentum scattering. We find
that transport in the (001) 2DEG is dominated by a single pair of bands, weakly
coupled by impurity scattering, one of which has a larger Fermi velocity while
the other provides an efficient current-relaxation mechanism. This leads to
strong angle-dependent current damping and a large AMR with many angular
harmonics. In contrast, AMR in the (111) 2DEG typically features a single
$\cos(2\vartheta)$ harmonic, with the angle-averaged magnetoresistance being
highly tunable by a symmetry-allowed trigonal distortion. We also explore how
the (111) 2DEG Fermi sufaces are impacted by electronic nematicity via a
surface phase transition into a 2D polar metal for which we discuss a Landau
theory, and show that this leads to distinct symmetry components and higher
angular harmonics in the AMR. Our results are in qualitative agreement with
experiments from various groups for 2DEGs at the SrTiO$_3$ surface or the
LaAlO$_3$-SrTiO$_3$ interface.
