We investigate nonlinear transport signatures stemming from linear and cubic spin-orbit interactions in one- and two-dimensional systems. The analytical zero-temperature response to external fields is complemented by a finite-temperature numerical analysis, establishing a way to distinguish between linear and cubic spin-orbit interactions. We also propose a protocol to determine the relevant material parameters from transport measurements attainable in realistic conditions, illustrated by values for Ge heterostructures. Our results establish a method for the fast benchmarking of spin-orbit properties in semiconductor nanostructures.