A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) variety of regular semigroups. For an (e)-variety (\mathscr{V}) of locally inverse or (E)-solid regular semigroups, the bifree object (B F \mathcal{Y}(X)) on a set (X) is the natural concept of a "free object" in (\mathcal{Y}). Its existence has been proved by (Y). T. Yeh. In this paper, the bifree locally inverse semigroup (B F \mathscr{L} \mathscr{I}(X)) is described as a homomorphic image of the absolutely free algebra of type (\langle 2,2\rangle) generated by (X) and the set of formal inverses (X^{\prime}), and equivalently as subsemigroup of a semidirect product of a suitable free semilattice by the bifree completely simple semigroup on (X). This latter realization is used to show that (B F \mathscr{\mathscr { I }}(X)) is combinatorial, completely semisimple and satisfies several finiteness conditions. Furthermore, the approach of biidentities is used to formulate a Birkhoff-type theorem for (e)-varieties of locally inverse semigroups and to establish a one-one correspondence between locally inverse (e)-varieties and fully invariant congruences on (B F \mathscr{\mathscr { I }}(X)) for countably infinite (X). As an application, it is shown that in each e-variety of locally inverse semigroups all free products exist. (1994 Academic Press, Inc.