  The 3 X 3 Majorana neutrino mass matrix is written as a sum of 3 terms, i.e.
M_nu = M_A + M_B + M_C, where M_A is proportional to the identity matrix and
M_B and M_C are invariant under different Z_3 transformations. This M_nu is
very suitable for understanding atmospheric and solar neutrino oscillations,
with sin^2 (2 theta_atm) and tan^2 (theta_sol) fixed at 1 and 0.5 respectively,
in excellent agreement with present data. It has in fact been proposed before,
but only as an ansatz. This paper uncovers its underlying symmetry, thus
allowing a complete theory of leptons and quarks to be constructed.
