  In this paper we continue work in the direction of a characterization of
rational period functions on the Hecke groups. We examine the role that
Hecke-symmetry of poles plays in this setting, and pay particular attention to
non-symmetric irreducible systems of poles for a rational period function. This
gives us a new expression for a class of rational period functions of any
positive even integer weight on the Hecke groups. We illustrate these
properties with examples of specific rational period functions. We also correct
the wording of a theorem from an earlier paper.
