  We describe a qualitative feature of the AdS_5 x S^5 string spectrum which is
not captured by the asymptotic Bethe ansatz. This is reflected by an enhanced
discrete symmetry in the asymptotic limit, whereby extra energy degeneracy
enters the spectrum. We discuss how finite size corrections should lift this
degeneracy, through both perturbative (Luscher) and non-perturbative
approaches (the Mirror TBA), and illustrate this explicitly on two such
asymptotically degenerate states.
