Semigroups of transformations preserving an equivalence relation and a cross-section

For a set X, an equivalence relation rho on X, and a cross-section R of the partition X/rho induced by rho, consider the semigroup T(X, rho, R) consisting of all mappings a from X to X such that a preserves both rho (if (x, y) is an element of rho then (xa, ya) is an element of rho) and R (if r is an element of R then ra is an element of R). The semigroup T(X, rho, R) is the centralizer of the idempotent transformation with kernel rho and image R. We determine the structure of T(X, rho, R) in terms of Green's relations, describe the regular elements of T(X, rho, R), and determine the following classes of the semigroups T(X, rho, R); regular, abundant, inverse, and completely regular.