I think you may have seen my golf just before I posted a fix. The bitwise OR can't be used here in this way, since it has the same precedence as XOR, and this leads to a rather subtle bug (notice that your sub returns 26 as a symmetrical number). But your idea still applies; here's a working one at 56:

  +{map$_^reverse||reverse($x=$_**2)^$x?():($_,$x),0..pop}
[download]

update: Switching from XOR to subtraction, however, permits the use a bitwise OR, saving a char, at 55:

  +{map$_-reverse|reverse($x=$_**2)-$x?():($_,$x),0..pop}
[download]

   MeowChow                                   
               s aamecha.s a..a\u$&owag.print