We calculate the nonlinear cotunneling conductance through interacting quantum-dot systems in the deep Coulomb blockade regime using a rate equation approach based on the T-matrix formalism, which shows in the concerned regions very good agreement with a generalized master equation approach. Our focus is on inelastic cotunneling in systems with weakly broken degeneracies, such as complex quantum dots or molecules. We find for these systems a characteristic gate dependence of the nonequilibrium cotunneling conductance. While on one side of a Coulomb diamond the conductance decreases after the inelastic cotunneling threshold toward its saturation value, on the other side it increases monotonously even after the threshold. We show that this behavior originates from an asymmetric gate voltage dependence of the effective cotunneling amplitudes.