Example for an undecidable language L such that L is reducible to its complement and vice versa
Let $K$ be some undecidable problem, and define
$$
L = \{0x : x \in K \} \cup \{1y : y \notin K \}.
$$
(I'm assuming the alphabet is $\{0,1\}$.)
Then
$$
\overline{L} = \{\epsilon\} \cup \{0x : x \notin K \} \cup \{1y : y \in K \}.
$$
I'll let you take it from here.