I'm trying to prove the undecidability of the following language.
$$L=\{\langle M \rangle\mid M\text{ is a Turing machine and there is a string }w\\\text{ s.t. }M\text{ accepts }w\text{ and }M\text{ rejects }w'\},$$ where $w'$ is the mirrored version of $w$.
I know that my first steps should be to find a reduction from $A_{\text{TM}}$, which is undecidable but the rejection part of the mirrored string is proving troublesome.