This paper contains deep results in the theory of existence varieties of regular semigroups. First of all, a suitable equational logic for e-varieties of \(E\)-solid semigroups is developed which extends the one for orthodox semigroups presented by the same authors [in Semigroup Forum 40, No. 3, 257-296 (1990; Zbl 0705.20052)]. The approach is as follows. Let \(X\) be a countably infinite set of variables, \(X'=\{x'\mid x\in X\}\) be a disjoint copy of \(X\) and let \(U'(X)\) be the free unary semigroup on \(X\cup X'\), the unary operation being denoted by \(^{-1}\). Let \(G(X)\) be the free group on \(X\) and \(K(X)\) be the kernel of the morphism \(U'(X)\to G(X)\) determined by \(x\mapsto x\), \(x'\mapsto x^{-1}\). Moreover, let \({F'}^\infty(X)\) be the smallest subsemigroup of \(U'(X)\) containing \(X\cup X'\) and being closed under the partial unary operation assigning \((u)^{-1}\) to every word \(u\in K(X)\). Let \(S\) be an \(E\)-solid semigroup and let \(\theta\colon X\cup X'\to S\) be a matched mapping (that is, \(x\theta\) and \(x'\theta\) are mutual inverses of each other). Then \(\theta\) extends uniquely to a morphism \(\Theta\colon{F'}^\infty\to S\). A formal equality \(u\simeq v\) (\(u,v\in{F'}^\infty\)) is a bi-identity and an \(E\)-solid semigroup \(S\) is said to satisfy this bi-identity if \(u\Theta=v\Theta\) for each matched mapping \(\theta\). Subject to this notion of identity the classical results on varieties of universal algebras carry over to the context of e-varieties of \(E\)-solid semigroups: there are analogues of free objects (bifree objects), the set of all e-varieties of \(E\)-solid semigroups forms a complete lattice under inclusion which is antiisomorphic to the lattice of all fully invariant congruences on the bifree \(E\)-solid semigroup on an infinite set \(X\) (and also to the lattice of (bi)equational theories), etc. In addition it is shown that the concept of bi-identity can be simplified when dealing with \(E\)-solid locally orthodox semigroups. Sections 3-8 of the paper present several applications, the most important of which yield concrete descriptions of the bifree objects in certain \(E\)-solid locally orthodox e-varieties. Let \(\mathcal Q\) be an e-variety of completely simple semigroups containing all rectangular bands, let \(\mathcal K\) be a group variety being contained in \(\mathcal Q\). Let \(\mathcal Y\) be a band monoid variety and let \(\mathcal P(\mathcal Y)\) be the variety of bands all of whose local submonoids belong to \(\mathcal Y\). For a class \(\mathcal C\) of regular semigroups denote by \(\langle\mathcal C\rangle\) the e-variety generated by \(\mathcal C\). The paper mainly deals with e-varieties of the form \(\langle\mathcal P(\mathcal Y)\circ\mathcal Q\cap\mathcal{CR}\circ\mathcal K\rangle\) where \(\circ\) denotes the Mal'cev product (within the class of regular semigroups) and \(\mathcal{CR}\) is the class of all completely regular semigroups. Each of these classes is \(E\)-solid and locally orthodox. What is more, for the special cases \(\mathcal Q=\) all completely simple semigroups and \(\mathcal K=\) all groups we get the class of all \(E\)-solid locally orthodox semigroups with bands in the local submonoids being in \(\mathcal Y\). A concrete description of the bifree objects on \(X\) in these aforementioned classes is given (modulo bifree objects in \(\mathcal Q\) and free groups in \(\mathcal K\)) by embedding these objects into precisely described Pastijn products of some bifree band in \(\mathcal P(\mathcal Y)\) (on an enlarged set of generators) by the bifree object on \(X\) in \(\mathcal Q\). The Pastijn product is a construction very reminiscing of a semidirect product which has been introduced by \textit{J. Kaďourek} [Int. J. Algebra Comput 6, No. 6, 761-788 (1996; Zbl 0880.20041)] and actually goes back to \textit{F. Pastijn} [Trans. Am. Math. Soc. 273, 631-655 (1982; Zbl 0512.20042)]. A former version of this important paper has already appeared in Semigroup Forum 58, No. 1, 17-68 (1999; see the preceding note Zbl 0931.20048). Unfortunately, that version contains a lot of misprints that have been introduced during the editorial process apparently caused by the incompatibility of various versions of \TeX. These errors are disturbing enough to destroy the readability of the paper.