A by now standard way of studying the lattice of congruences on a regular semigroup is via various operators. This approach originated with the study of the \(T\) and \(K\) operators on an inverse semigroup: with any congruence \(\rho\) are associated \(\rho k\), \(\rho K\), \(\rho t\) and \(\rho T\), respectively the least and greatest congruences having the same kernel (union of idempotent classes) as \(\rho\), and the least and greatest congruences having the same trace (restriction to the idempotents) as \(\rho\). These operators generalize to regular semigroups in several ways. In this paper, the \(T\) operator is studied in conjunction with the \(V\) operator: here \(\rho V\) is the largest congruence such that each idempotent class of \(\rho V/\rho\) is a rectangular band; and \(\rho v\) is the smallest congruence such that \(\rho/\rho v\) has that property. This operator was originally studied by \textit{F. Pastijn} and \textit{M. Petrich} [J. Pure Appl. Algebra 53, No. 1/2, 93-123 (1988; Zbl 0649.20052)]. Regarded as operators on the lattice of congruences, \(v,V,t,T\) generate a semigroup, the ``\(VT\)-operator semigroup''. Via some elegant arguments, it is shown that for any band of groups (or ``cryptogroup''), this operator semigroup is a quotient of a certain explicitly constructed seventeen-element semigroup. Since a completely simple semigroup is already known with exactly this operator semigroup, it is therefore the most general one. (Note that for completely simple semigroups, \(V=K\).) For orthogroups (orthodox completely regular semigroups), a similar theorem is proven. However it relies on a detailed analysis of these operators on free regular orthogroups to produce an example analogous to the completely simple semigroup cited above.