Eh? I got |00> + |11> = |LL> + |RR> as expected (with correct coefficients) in your case (as well as, I bet, in mine).
That's what I was getting at with my "Hadamard Basis" nonsense, but it shouldn't have taken this "flash of insight" (or any calculation) to see why the statement should be true.
EDIT: This only shows an explicit change of basis that works when all the constants xi are the same in my original state. I haven't figured out how to think about it in general.