  A (discrete) dynamical system may have various symmetries and reversing
symmetries, which together form its so-called reversing symmetry group. We
study the set of 3D trace maps (obtained from two-letter substitution rules)
which preserve the Fricke-Vogt invariant I(x,y,z). This set of dynamical
systems forms a group G isomorphic with the projective linear (or modular)
group PGL(2,Z). For such trace maps, we give a complete characterization of the
reversing symmetry group as a subgroup of the group A of all polynomial
mappings that preserve I(x,y,z).
