If $A \leq B$, $B \leq \overline{A}$, and $B$ is r.e., is $A$ recursive?
Notice that $A \leq B \leq \overline{A}$. Since $B$ is r.e., $A$ is r.e., hence $\overline{A}$ is co-r.e. Since $A \leq \overline{A}$, $A$ is also co-r.e. Hence $A$ is recursive.