  Current and upcoming experiments should improve the sensitivity to radiative
lepton decays by an order of magnitude. This paper assumes that one of the tau
--> (e, mu) gamma decays is observed, and explores the structure and
consequences of the required new flavoured couplings. In simple models (a
low-scale seesaw, leptoquarks) it is shown that the dipole vertex function is
proportional to a product of flavoured matrices from the Lagrangian (a
"Jarlskog-like" invariant), provided that the loop particles are weakly coupled
to the Higgs. Secondly, if the dipole vertex function has a hierarchical
structure, this can imply that only some of the tau --> (e, mu) gamma modes can
be observed, due to the "approximate zero" implied by the bound on mu --> e
gamma. The assumptions underlying this potential test of a hierarchical
structure are discussed.
