I'm given the set $T = \{\langle M, w\rangle : M $ is a Turing Machine that accepts $w^\mathcal R$ whenever it accepts $w \}$ and I want to show it's undecidable but recognizable. (I'm using the bracket notation to denote the encoding of the TM and input string)
Given that I know the set $A = \{\langle M,w\rangle : M \text{ accepts } w \}$ is undecidable but recognizable, I'm trying to show that $A$ reduces to $T$. I start by assuming that $T$ is decided by a Turing Machine $R$ and then attempt to construct a new Turing Machine, $S$, that will decide $A$ by using $R$ as a subroutine.
To construct this new machine $S$, I will run $R$ on input $\langle M,w\rangle$. If $R$ accepts, then $\langle M,w\rangle$ is in $A$. However, $R$ rejecting doesn't necessarily mean that $\langle M,w\rangle$ isn't in $A$, since $M$ might reject $w$ reversed but accept $w$. So now I'm confused on how to actually construct the reduction. Any help would be great