We report on the detailed study of multicomponent spin waves in an s = 3/2 Fermi gas where the high spin leads to novel tensorial degrees of freedom compared to s = 1/2 systems. The excitations of a spin-nematic state are investigated from the linear to the nonlinear regime, where the tensorial character is particularly pronounced. By tuning the initial state we engineer the tensorial spin-wave character, such that the magnitude and the sign of the counterflow spin currents are effectively controlled. A comparison of our data with numerical and analytical results shows good agreement.