Proving that it is undecidable if a Turing machine accepts a language that is its own reverse I have a Turing machine M. How can I prove that $L(M) = (L(M))^R$ is decidable by constructing a Turing machine that can do this? I know how to figure out if two DFA's accept the same language (using the symmetric difference) however I don't believe it works for Turing machines. Now I am able to design a Turing machine where, for a string w, I can determine whether or not the reverse of w is also accepted by the language. The problem is that I'm having trouble determining if the set of all strings is the same as the set of all reverse strings. If I design another TM then I feel like it's just going to infinitely loop. So how can I prove that this is decidable?
The decision problem


  Given Turing machine $M$, is $L(M) = (L(M))^R$?


is undecidable. This is a simple consequence of Rice's theorem.

The property (i.e. class of Turing-recognizable languages)

$$\mathcal{S}_{\mathrm{rev}} = \{ L \mid L \; \text{recognizable and } L = L^R \}$$

is non-trivial, since $\emptyset \in \mathcal{S}_{\mathrm{rev}}$ and $\{ ab \} \notin \mathcal{S}_{\mathrm{rev}}$.