The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\Sigma_1)$, then $$Z(M_U) = Z(M_{-U})$$ where $-I\in \mathrm{Mod}(\Sigma_1)$ is the hyperelliptic involution.

What about the genus 2 case? If $-I\in \mathrm{Mod}(\Sigma_2)$ is the hyperelliptic involution on the genus 2 surface, then is there any $U\in \mathrm{Mod}(\Sigma_2)$, for which the following does not hold? $$Z(M_U) \stackrel{?}{=} Z(M_{-U})$$