 Ambivalent group

A finite group is ambivalent when every element is conjugate to its inverse. Since every complex character satisfies $\chi (g^{-1})=\overline{\chi (g)}$, an ambivalent group has real values for every irreducible character. Conversely, if every irreducible value is real, irreducible characters separate conjugacy classes and force every $g$ to be conjugate to $g^{-1}$.