  An involutive diffeomorphism $\sigma$ of a connected smooth manifold $M$ is
called dissecting if the complement of its fixed point set is not connected.
Dissecting involutions on a complete Riemannian manifold are closely related to
constructive quantum field theory through the work of Dimock and Jaffe/Ritter
on the construction of reflection positive Hilbert spaces. In this article we
classify all pairs $(M,\sigma)$, where $M$ is an irreducible symmetric space,
not necessarily Riemannian, and $\sigma$ is a dissecting involutive
automorphism. In particular, we show that the only irreducible $1$-connected
Riemannian symmetric spaces are $S^n$ and $H^n$ with dissecting isometric
involutions whose fixed point spaces are $S^{n-1}$ and $H^{n-1}$, respectively.
