Symbolic dynamics for the piecewise rotations: Case of the bijective symmetric maps

We consider a specific %piecewise rotation of the plane that is continuous on two half-planes, class of piecewise rotations of the plane that are continuous on two half-planes, as studied in \cite{Bosh.Goet.03}, \cite{Goet.Quas.09} and \cite{Che.Goe.Qua.12}. %If Assuming that the angle belongs to the set $\{\frac{\pi}{2},\frac{\pi}{3},\frac{\pi}{6},\frac{\pi}{4}\}$ $, we give a description of the symbolic dynamics of this map in the bijective symmetric case.