A regular semigroup is fundamental [E-disjunctive] if it supports no nontrivial idempotent-separating [idempotent-pure] congruence. Every regular semigroup is a subdirect product of a fundamental semigroup and an E-disjunctive one. Denote by \({\mathcal L}({\mathcal C}{\mathcal R})\) the lattice of varieties of completely regular semigroups and by \({\mathcal F}\) and \({\mathcal D}\) the classes of fundamental and of E-disjunctive completely regular semigroups, respectively. This paper studies, in the main, the maps that associate with each variety \({\mathcal V}\) of completely regular semigroups the classes \({\mathcal V}\cap {\mathcal F}\) and \({\mathcal V}\cap {\mathcal D}\). The ranges form complete lattices (but not sublattices of \({\mathcal L}({\mathcal C}{\mathcal R}))\); the map \({\mathcal V}\to {\mathcal V}\cap {\mathcal F}\) is in fact a complete homomorphism, which induces the well-known congruence “T” on \({\mathcal L}({\mathcal C}{\mathcal R})\) (see, for example [F. Pastijn, J. Aust. Math. Soc., Ser. A 49, 24-42 (1990; see the preceding review Zbl 0706.20042)]). However, the map \({\mathcal V}\to {\mathcal V}\cap {\mathcal D}\) behaves less satisfactorily: it is a complete meet- homomorphism, but does not respect joins. The relationship between the induced partition and the well-known “K” congruence on \({\mathcal L}({\mathcal C}{\mathcal R})\) is explored.