Determining whether the language of a DFA is closed under reversal
For a language $L$, we define the reverse language $L^{\mathcal{R}}$ by
$$ L^{\mathcal{R}} = \{ x^{\mathcal{R}} : x \in L \}. $$
In words, $L^{\mathcal{R}}$ is obtained from $L$ by reversing all words.

It follows that $L = L^{\mathcal{R}}$ if and only if, for each word $x$, we have $x \in L \Leftrightarrow x^{\mathcal{R}} \in L$.