Show that the following language is not recursive: $L = \{w \mid M_w \text{exists and it accepts a word } x_1 = 0x \text{ if and only if it accepts } x_2 = 1x\}$ ($x \in \Sigma^*$, so $x_1$ is starting with a 0 and $x_2$ is obtained from $x_1$ by changing the first 0 to a 1).
We say a language $L$ is good if "$0x\in L$ if and only if $1x\in L$". Let $\mathcal{L}$ be the set of all good languages. Then you are trying to determine whether the language recognized by a given TM belongs to $\mathcal{L}$. Since $\mathcal{L}$ is non-trivial, according to Rice's theorem, this is undecidable.

If you want a proof without Rice's theorem, you can take a look at the proof of Rice's theorem and try to write a proof for this specific problem. This is not hard.