Proving that it is undecidable if a Turing machine accepts a language that is its own reverse
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 $\{ ab \} \notin \mathcal{S}_{\mathrm{rev}}$.