Let \(S\) be a completely regular semigroup. The (right) Schützenberger representation of \(S\) is the morphism \(\rho: S \to {\mathcal P\mathcal T}(S)\), where for \(a \in S\), \(\rho_ a\) denotes right translation by \(a\), restricted to the principal ideal \(S a S\) that it generates. (Here \({\mathcal P\mathcal T}(X)\) denotes the semigroup of partial transformations on a set \(X\).) The product of this representation with its left-right dual is faithful. Observe that for each \(a \in S\), the domain of \(\rho_ a\) is a union of \(\mathcal D\)-classes and for every \(\mathcal D\)-class \(\alpha\) contained in \(S a S\), \(\rho_ a\) induces a full transformation \(\rho_{a,\alpha}\) of \(\alpha/{\mathcal L}\) (by Green's Lemmas). This `right regular representation' of \(S\) is used as a model to construct `natural' completely regular semigroups of partial transformations. Let \(X\) be any nonempty set and let \(\lambda\), \(\delta\) be equivalence relations on \(X\) such that \(\lambda \subseteq \delta\) and the \(\lambda\)-classes within any given \(\delta\)-class all have the same cardinality. For such an `\(r\)-triple' \((X,\lambda,\delta)\), \({\mathcal T}(X,\lambda,\delta)\) denotes the set of partial transformations \(\phi\) of \(X\) such that \((i)\) the domain of \(\phi\) is a union of \(\delta\)- classes, (ii) \(x \delta x\phi\) and (iii) \(\phi\) induces a bijection of \(x \lambda\) on \((x\phi)\lambda\), whenever \(x \phi\) is defined; \({\mathcal T}(X,\lambda,\delta)\) is a regular subsemigroup of \({\mathcal P}{\mathcal T}(S)\). A `cr-system' \((X,\lambda,\delta,\{U_ \alpha\})\) is an \(r\)-triple \((X,\lambda,\delta)\) together with, for each \(\delta\)-class \(\alpha\) of \(S\), a completely regular subsemigroup \(U_ \alpha\) of the full transformation semigroup \({\mathcal T}(\alpha/\lambda)\). For any such system, \({\mathcal T}(X,\lambda,\delta,\{U_ \alpha\})\) denotes the set of \(\phi \in {\mathcal T}(X,\lambda,\delta)\) such that for each \(\delta\)-class \(\alpha\) contained in the domain of \(\phi\), the transformation of \(\alpha/\lambda\) induced by \(\phi\) belongs to \(U_ \alpha\). Then \({\mathcal T}(X,\lambda,\{U_ \alpha\})\) is a completely regular subsemigroup of \({\mathcal P\mathcal T}(X)\). It is shown that if each \(U_ \alpha\) belongs to some variety \(\mathcal U\) of completely regular semigroups, then \({\mathcal T}(X,\lambda,\delta,\{U_ \alpha\})\) belongs to the Malcev product \({\mathcal L}G \circ {\mathcal V}\), where \({\mathcal L}G\) is the variety of left groups. Various examples of the construction are given. In the notation of the first paragraph, the right regular representation maps \(S\) into \({\mathcal T}(X,{\mathcal L},{\mathcal D},\) \(\{U_ \alpha\})\), where for each \(\mathcal D\)-class \(\alpha\), \(U_ \alpha = \{\rho_{a,\alpha}: \alpha \subset SaS\}\). The scope of the construction is demonstrated by the fact that any free object in \({\mathcal L}G \circ {\mathcal V}\) is embeddable in such a semigroup. Some special cases are considered. For instance, Petrich's representation of completely simple semigroups is an example. It is shown that in general \({\mathcal T}(X,\lambda,\delta,\{U_ \alpha\})\) can be decomposed as a strong semilattice of completely regular semigroups, each of which is a direct product of wreath products of a full symmetric group and a completely regular transformation semigroup.