  It is well-known that, in the basis where the charged-lepton mass matrix is
diagonal, there are seven cases of two texture zeros in Majorana neutrino mass
matrices that are compatible with all experimental data. We show that two of
these cases, namely B3 and B4 in the classification of Frampton, Glashow and
Marfatia, are special in the sense that they automatically lead to near-maximal
atmospheric neutrino mixing in the limit of a quasi-degenerate neutrino mass
spectrum. This property holds true irrespective of the values of the solar and
reactor mixing angles because, for these two cases, in the limit of a
quasi-degenerate spectrum, the second and third row of the lepton mixing matrix
are, up to signs, approximately complex-conjugate to each other. Moreover, in
the same limit the aforementioned cases also develop a maximal CP-violating
CKM-type phase, provided the reactor mixing angle is not too small.
