This is related to a homework question for a class that I am TAing for. I'm using Sipser terminology here (recognizable for computably enumerable, decidable for computable).
Given that $w^r$ is the reverse of $w$, and that $T = \{ \langle M\rangle \mid \mbox{if $M$ accepts $w$ then $M$ accepts $w^r$}\}.$
Now here are the questions:
Clearly $T$ is undecidable. Is $T$ unrecognizable? My intuition tells me yes it's unrecongnizable. Since we have to confirm that $w^r$ is accepted for all $w$, this seems to be a good sign that it is unrecognizable.
Is $\overline{T}$ recognizable? This also seems unrecognizable as we would have no way to tell if the machine is looping on a particular input.