  Contrary to the quark mixing matrix, the lepton mixing matrix could be
symmetric. We study the phenomenological consequences of this possibility. In
particular, we find that symmetry would imply that |U_{e3}| is larger than
0.16, i.e., above its current 2 sigma limit. The other mixing angles are also
constrained and CP violating effects in neutrino oscillations are suppressed,
even though |U_{e3}| is sizable. Maximal atmospheric mixing is only allowed if
the other observables are outside their current 3 sigma ranges, and sin^2
theta_{23} lies typically below 0.5. The Majorana phases are not affected, but
the implied values of the solar neutrino mixing angle have some effect on the
predictions for neutrinoless double beta decay. We further discuss some formal
properties of a symmetric mixing matrix.
