  To address the issue of whether tri-bimaximal mixing (TBM) is a softly-broken
hidden or an accidental symmetry, we adopt a model-independent analysis in
which we perturb a neutrino mass matrix leading to TBM in the most general way
but leave the three texture zeros of the diagonal charged lepton mass matrix
unperturbed. We compare predictions for the perturbed neutrino TBM parameters
with those obtained from typical SO(10) grand unified theories with a variety
of flavor symmetries. Whereas SO(10) GUTs almost always predict a normal mass
hierarchy for the light neutrinos, TBM has a priori no preference for neutrino
masses. We find, in particular for the latter, that the value of |U_{e3}| is
very sensitive to the neutrino mass scale and ordering. Observation of
|U_{e3}|^2 > 0.001 to 0.01 within the next few years would be incompatible with
softly-broken TBM and a normal mass hierarchy and would suggest that the
apparent TBM symmetry is an accidental symmetry instead. No such conclusions
can be drawn for the inverted and quasi-degenerate hierarchy spectra.
