  Two sets of four 3x3 matrices 1^(3), varphi_1, varphi_2, varphi_3 and 1^(3),
mu_1, mu_2, mu_3 are constructed, forming two unitarily isomorphic reducible
representations 3 of the group Z_2 x Z_2 called often the four-group. They are
related to each other through the effective neutrino mixing matrix U with
s_{13} = 0, and generate four discrete transformations of flavor and mass
active neutrinos, respectively. If and only if s_{13} = 0, the generic form of
effective neutrino mass matrix M becomes invariant under the subgroup Z_2 of
Z_2 x Z_2 represented by the matrices 1^(3) and varphi_3. In the approximation
of m_1 = m_2, the matrix M becomes invariant under the whole Z_2 x Z_2
represented by the matrices 1^(3), varphi_1, varphi_2, varphi_3. The effective
neutrino mixing matrix U with s_{13} = 0 is always invariant under the whole
Z_2 x Z_2 represented in two ways, by the matrices 1^(3), varphi_1, varphi_2,
varphi_3 and 1^(3), mu_1, mu_2, mu_3.
