Inhomogeneous broadening has the potential of introducing a new picosecond-range quasi-relaxation rate in an electronically twofold-degenerate system such as a molecule whose symmetry is square (D-4h). The physics of the process is the mutual cancellation of oscillatory contributions whose periods are determined by the random energy splittings. The effect occurs only for quite specific conditions of excitation and line broadening, and may be useful in the interpretation of existing experiments on magnesium tetraphenyl porphyrin. (C) 2002 Elsevier Science B.V. All rights reserved.