Reversible finite automata with halting states (RFA) were first considered by Ambainis and Freivalds to facilitate the research of KondacsWatrous quantum finite automata. In this paper we consider some of the algebraic properties of RFA, namely the varieties these automata generate. Consequently, we obtain a characterization of the Boolean closure of the classes of languages recognized by these models. We also obtain an equality which relates varieties of ordered J -trivial monoids with the variety of R-trivial monoids.