Undecidability closed

Question

REVERSE = { M | M is a TM with the property: for all w, M(w) accepts iff M(w^R) accepts}

Here getting confused for applying condition to apply Rice's theorem for the given property 

T_yes={Palindromes}    T_no={Non-palindromes} 

Because of which it is non-trivial property and hence Turing Undecidable(according to Rice's theorem)

Is my analysis above is correct here ?

and is it Turing Recognisable ?can we apply Rice's 2nd theorem?

Source:https://goo.gl/UIhNyT  (Problem 1)