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We continue the study of permutations of a finite regular semigroup that mapeach element to one of its inverses, providing a complete description in thecase of semigroups whose idempotent generated subsemigroup is a union ofgroups. We show, in two ways, how to construct an involution matching on thesemigroup of all transformations which either preserve or reverse orientationof a finite cycle. Finally, by way of application, we prove that finite fulltransformation semigroups have no cover by inverse semigroups that is closedunder intersection.

Involution matchings, the semigroup of orientation-preserving and  orientation-reversing mappings, and inverse covers of the full transformation  semigroup