  We study natural lepton mass matrices, obtained assuming the stability of
physical flavour observables with respect to the variations of individual
matrix elements. We identify all four possible stable neutrino textures from
algebraic conditions on their entries. Two of them turn out to be uniquely
associated to specific neutrino mass patterns. We then concentrate on the
semi-degenerate pattern, corresponding to an overall neutrino mass scale within
the reach of future experiments. In this context we show that i) the neutrino
and charged lepton mixings and mass matrices are largely constrained by the
requirement of stability, ii) naturalness considerations give a mild preference
for the Majorana phase most relevant for neutrinoless double-beta decay,
$\alpha \sim \pi/2$, and iii) SU(5) unification allows to extend the
implications of stability to the down quark sector. The above considerations
would benefit from an experimental determination of the PMNS ratio
$|U_{32}/U_{31}|$, i.e. of the Dirac phase $\delta$.
