G. Trotter and the author introduced a "regular" semidirect product U * V of e-varieties U and V. Among several specific situations investigated there was the case V = RZ, the e-variety of right zero semigroups. Applying a covering theorem of McAlister, it was shown there that in several important cases (for instance for the e-variety of inverse semigroups), U * RZ is precisely the e-variety LU of "locally U" semigroups.The main result of the current paper characterizes membership of a regular semigroup S in U * RZ in a number of ways; one in terms of an associated category S E and another in terms of S regularly dividing a regular Rees matrix semigroup over a member of U. The categorical condition leads directly to a characterization of the equality U * RZ = LU in terms of a graphical condition on U, slightly weaker than "e-locality." Among consequences of known results on e-locality, the conjecture CR * RZ = LCR (with CR denoting the e-variety of completely regular semigroups), is therefore verified. The connection with matrix semigroups then leads to a range of Rees matrix covering theorems that, while slightly weaker than McAlister's, apply to a broader range of examples. K. Auinger and P. G. Trotter (Pseudovarieties, regular semigroups and semidirect products, J. London Math. Soc. (2) 58 (1998), 284-296) have used our results to describe the pseudovarieties generated by several important classes of (finite) regular semigroups.An e-variety of regular semigroups is a class of regular semigroups that is closed under products, quotients, and regular subsemigroups. In a recent paper [6], Trotter and the author introduced a product U * V of e-varieties U V, well-defined if (and only if) either U or V consists of completely simple semigroups, as follows: U * V is the e-variety generated by the semigroups of regular elements of the wreath products (or the semidirect prod-287