  We present and verify experimentally a universal theoretical framework for
the description of spin-charge interconversion in non-magnetic metal/insulator
structures with interfacial spin-orbit coupling (ISOC). Our formulation is
based on drift-diffusion equations supplemented with generalized boundary
conditions. The latter encode the effects of ISOC and relate the electronic
transport in such systems to spin loss and spin-charge interconversion at the
interface, which are parameterized, respectively, by $G_{\parallel/\perp}$ and
$\sigma_{\rm{sc/cs}}$. We demonstrate that the conversion efficiency depends
solely on these interfacial parameters. We apply our formalism to two typical
spintronic devices that exploit ISOC: a lateral spin valve and a multilayer
Hall bar, for which we calculate the non-local resistance and the spin Hall
magnetoresistance, respectively. Finally, we perform measurements on these two
devices with a BiO$_x$/Cu interface and verify that transport properties
related to the ISOC are quantified by the same set of interfacial parameters.
