  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.
