The coupling of spin and electronic degrees of freedom via the spin-orbit interaction (SOI) is an essential ingredient for many proposed future technologies. However, probing the strength and nature of SOI is a significant challenge, especially in heterostructures. Here, we consider the nonlinear Shubnikov-de Haas (NSdH) effect, a quantum oscillatory effect that occurs under conditions similar to those of the well-known SdH effect, but is second order in the applied electric field. We demonstrate that, unlike its linear counterpart, the NSdH effect is highly sensitive to the spin textures that arise from SOI. We show that the relative phases and beating patterns of the nonlinear oscillations provide additional information that discriminates linear- and cubic-Rashba-dominated regimes in the models considered here. Our results establish NSdH as a complementary phase-resolved probe of SOI, offering a new framework for characterizing materials relevant to topology, spintronics, and solid-state quantum information technologies.