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.
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