Hypermaps over non-abelian simple groups and strongly symmetric generating sets

A generating pair $x,y$ for a group $G$ is said to be $\mathit{\text{symmetric}}$ if there exists an automorphism ${\phi }_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{{\phi }_{x,y}}=x^{-1}$ and $y^{{\phi }_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be $\mathit{\text{strongly}}\mathit{\text{symmetric}}$ if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.