In recent years, large amplitude oscillatory shear (LAOS) has become a prime tool to investigate the nonlinear rheology of complex fluids. More specifically, most studies use LAOS data as a macroscopic probe for hypothetical microscopic changes, sometimes successfully. Nevertheless, we would like to raise awareness on the potential impact of secondary flows triggered by instabilities in LAOS performed in curved shear flows exemplified by the Taylor-Couette geometry. First, we show that even for Newtonian fluids, where no micro-structural changes are expected, complex flow patterns can emerge in the LAOS data. Second, we stress the potential impact of similar effects in complex fluids by studying LAOS flows of a well-known shear-banding surfactant solution. Our hope is that our thorough study of the Newtonian case together with preliminary experiments on a complex fluid will reinforce the already successful analogy existing between inertial and elastic instabilities, suggest caution for future LAOS studies focused solely on a structural perspective, and open new research avenues combining flow instabilities and unsteady flows in complex fluids.