  Neutrino oscillation data indicate the presence of a 2-3 symmetry for the
left-handed neutrinos. We show that this symmetry between the second and third
generations cannot be extended to include the charged leptons, no matter what
basis we use. We point out that if this symmetry is also independently valid
for the right-handed neutrinos, then the active neutrino spectrum is inverted,
with $m_3=0$. This conclusion remains valid even when the left-handed 2-3
symmetry is broken by a non-maximal atmospheric mixing angle $\theta_{23}$, or
a non-zero reactor angle $\theta_{13}$. As previously shown by Mohapatra,
Nasri, and Yu, such a symmetry also gives rise to interesting consequences on
the leptogenesis asymmetry parameter $\epsilon_1$.
