Mirror quine (or my head hurts)
GolfScript, 46 chars

0{`".~#"+.-1%+\.!@@}.~##~.}@@!.\+%1-.+"#~."`{1


Well, this is ugly, but it works.  The output equals the code reversed, and is also a valid GolfScript program which outputs the original code again.

OK, let me try to explain how I constructed it.  First, I started from the quine {".~"}.~, and modified it as in this answer to reverse itself.  To make the output an executable quine in itself, I made a copy of the code before reversing it, and included a # at the end of the code, so that the reversed version at the end became just a comment.  Thus, we get the palindromic quine:

{`".~#"+.-1%}.~##~.}%1-.+"#~."`{


However, by the rules, palindromes are not allowed, so I needed to break the symmetry somehow.  I figured the easiest way would be to include a 0 (which, in itself, is a quine in GolfScript) in the code and flip it to 1 with ! after creating the reversed copy.  Most of the remaining complexity is just ugly stack manipulation to get everything in the right order.