  Strong Rashba spin-orbit coupling (SOC) of the two-dimensional electron gas
(2DEG) at the oxide interface $\mathrm{LaAlO_{3}/SrTiO_{3}}$ underlies a
variety of exotic physics, but its nature is still under debate. We derive an
effective Hamiltonian for the 2DEG at the oxide interface
$\mathrm{LaAlO_{3}/SrTiO_{3}}$ and find a different anisotropic Rashba SOC for
the $d_{xz}$ and $d_{yz}$ orbitals. This anisotropic Rashba SOC leads to
anisotropic static spin susceptibilities and also distinctive behavior of the
spin Hall conductivity. These unique spin responses may be used to determine
the nature of the Rashba SOC experimentally and shed light on the orbital
origin of the 2DEG.
