The main subject of this paper is the description of the congruences on certain monoids of transformations on a finite chain XnXn with n elements. Namely, we consider the monoids ORnORn and PORnPORn of all full, respectively partial, transformations on XnXn that preserve or reverse the orientation, as well as their respective submonoids OPnOPn and POPnPOPn of all orientation-preserving elements. The inverse monoid PORInPORIn of all injective elements of PORnPORn is also considered.