Let $M = \Gamma \backslash G/K$ be a Riemannian locally symmetric space, where $G$ is a connected semisimple Lie group of rank at least $2$, $K$ its maximal compact subgroup and $\Gamma < G$ an irreducible lattice.

**Question:** Does $M$ admit a global geodesic symmetry?

By a geodesic symmetry I mean an isometry of $M$ fixing a point $x\in M$ and reversing all geodesics through $x$. If the answer is NO then are there related general results?