  Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly
in systems whose dynamics are determined by a Lagrangian or Hamiltonian
function. For hybrid systems which possess a continuous-time dynamics
determined by a Lagrangian function, with a cyclic variable, the degrees of
freedom for the corresponding hybrid Lagrangian system can be reduced by means
of a method known as \textit{hybrid Routhian reduction}. In this paper we study
sufficient conditions for the existence of periodic orbits in hybrid Routhian
systems which also exhibit time-reversal symmetry. Likewise, we explore some
stability aspects of such orbits through the characterization of the
eigenvalues for the corresponding linearized Poincar\'e map. Finally, we apply
the results to find periodic solutions in underactuated hybrid Routhian control
systems.
