Orthodox semigroups and permutation matchings

We determine when an orthodox semigroup S has a permutation that sends each member of S to one of its inverses and show that if such a permutation exists, it may be taken to be an involution. In the case of a finite orthodox semigroup the condition is an effective one involving Green’s relations on the combinatorial images of the principal factors of S . We also characterise some classes of semigroups via their permutation matchings.