Cardinal Burke’s views on gender have been widely reported, including this:

Men’s behaviors and dress matter, for it affects how they relate to the world and it affects the culture. Men need to dress and act like men in a way that is respectful to themselves, to women and to children

To which, I concur with the following response at What Sister Never Knew and Father Never Told You 

While I agree with the statement, given His Eminence’s sartorial preferences I can only surmise that Cardinal Burke is about to transgender.  I wish him the best of luck in his new life.