  Recent experiments have shown that transition metal oxide heterostructures
such as SrTiO$_3$-based interfaces, exhibit large, gate tunable, spintronic
responses. Our theoretical study showcases key factors controlling the
magnitude of the conversion, measured by the inverse Edelstein and Spin Hall
effects, and their evolution with respect to an electrostatic doping. The
origin of the response can be linked to spin-orbital textures. These stem from
the broken inversion symmetry at the interface which produces an unusual form
of the interfacial spin-orbit coupling, provided a bulk atomic spin-orbit
contribution is present. The amplitudes and variations of these observables are
direct consequences of the multi-orbital subband structure of these materials,
featuring avoided and topological crossings. Interband contributions to the
coefficients lead to enhanced responses and non-monotonic evolution with
doping. We highlight these effects using analytical approaches and low energy
modeling.
