A symmetric measure-preserving system is one where the measure $Pr$ is preserved by two maps $T$ and $R$ where $R$ is self-inverse and $T\circ R = T$. We discuss the existence of such systems and some consequences, including when unimodal maps are conjugate to the symmetric tent map.
"Symmetric Measure-Preserving Systems." Real Anal. Exchange 24 (1) 411 - 422, 1998/1999.