I know that if you have a circuit $U$ that transforms $A → B$, it's possible to construct an inverse, ${U\dagger}(B) → A$. Is it also possible to transform the states with $T_{i,o}$ so that I can use the original circuit to do the reverse computation? Like so:
$$ U(T_o(B)) → T_i(A) $$
For example, I know the Toffoli gate is its own inverse. So $T_{i,o}$ can be the identity function:
$$U_{\operatorname{Tof}}(A) → B$$ $$U_{\operatorname{Tof}}(I(B)) → I(A)$$
I would like to know if some reasonable $T$ functions exists when $U$ is a universal circuit.
I'm a quantum computing newbie and coming at this question more from a physics standpoint, so not sure quite sure how to ask this most clearly. Suggestions are appreciated. Links to related research would also be great.
EDIT: $T$ may differ for input and output states.