Is there an undecidable language that is mapping reducible to its complement?
Yes, it is possible. Enumerate the set of all possible Turing machines and let $H$ (resp. $\overline{H}$) be the set of indices of the Turing machines that halt (resp. do not halt) on empty input.
Let $L = \{ \langle T, 1 \rangle \, : T \in H \} \cup \{ \langle T, 0 \rangle \, : T \in \overline{H} \}$.
Clearly $L$ is not decidable but it is possible to reduce $L$ to $\overline{L}$ since $\langle T, r \rangle \in L \iff \langle T, 1-r \rangle \in \overline{L}$.