Recent advances in on-surface chemistry, combined with scanning probe microscopy, have en abled the synthesis of correlated molecules on surfaces and the characterization of their chemical and electronic properties with unprecedented spatial resolution. Low-energy magnetic excitations of individual molecules are frequently investigated by scanning tunneling spectroscopy (STS) and often appear as symmetric step-like features in the differential conductance as a function of bias voltage. The interpretation of such steps is well established within cotunneling theory and effective model Hamiltonians (e.g., Hubbard- and spin-based models). Here, we extend the cotunneling formalism to general multireference systems. We show that multireference character, together with orbital dependent and strongly asymmetric tip/substrate couplings, can produce pronounced asymmetric line shapes in inelastic STS. These results provide an alternative microscopic mechanism for asym metric inelastic steps and peaks near the Fermi level frequently observed in STS experiments. In contrast to Kondo or Fano mechanisms, the asymmetry discussed here does not require a many-body resonance pinned to the Fermi level nor interference between a resonant and a non-resonant tunnel ing channel; it follows from the orbital composition of the neutral and virtually charged multiplets and can therefore appear at the bias of a finite-energy molecular excitation