Our theory of vibrationally inelastic collisions has previously been applied with success to atom-diatom collisions as well as to symmetric diatom-diatom collisions. It is based on an approximate correspondence between the classical and quantal equations of motion. Here we extend the theory to asymmetric diatom-diatom collisions. The results are in the form of analytic transition probabilities which depend on parameters calculated using exact classical trajectories. We compare our results with accurate quantal results reported for the collinear systems N 2 + CO, N 2 − OC. N 2 − O 2, H 2 + HBr, with harmonic-oscillator molecules and nearest-atom repulsive exponential interaction potentials. Comparisons are also made with calculations for the N 2 + N 2 mass combination, with, however, a varying ratio of oscillator frequencies, and with H 2 + D 2, in this case with an “MMBA” potential. Agreement is generally excellent for all transitions when the oscillators are near resonance (N 2 + CO, N 2 + OC). The agreement deteriorates as one considers systems further off resonance (N 2 − O 2) and with steeper potentials (H 2 + HBr). Serious errors, however, are made in only certain transitions. We present a detailed analysis, based on perturbation theory, in which the difficulty is traced to our neglect of double-jump operators generated by the bilinear term in the interaction potential. Inclusion of double jumps in the theory appears to be a major step which must await further work. Meanwhile, the theory is quantitative near resonance and still useful, with caution, even far off resonance.