An orthogroup is a union of groups in which the idempotents form a subsemigroup (orthodox union of groups). If in addition the idempotents form a regular band, the semigroup is a regular orthogroup. These semigroups form a variety when considered as semigroups with an inverse. The main result of the present paper is a description of the lattice of all subvarieties of this variety (modulo the variety of groups which is a subvariety). This is done by relating the fully invariant congruences on the free regular orthogroup to fully invariant congruences on free regular bands and free groups. The lattice is then studied in some detail. Various subdirect decompositions of it are obtained. A ''min-max'' network is described and bases for the identities in the various subvarieties is given.