Two sets of four 3\times 3 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 Z₂ \times Z₂ called often the four-group. They are related to each other through the effective neutrino mixing matrix U with s₁₃ = 0 and generate four discrete transformations of flavor and mass active neutrinos, respectively. If and only if s₁₃ = 0, the generic form of effective neutrino mass matrix M becomes invariant under the subgroup Z₂ of Z₂ \times Z₂ represented by the matrices {1}⁽³⁾ and \varphi ₃. In the approximation of m₁ = m₂, the matrix M becomes invariant under the whole Z₂ \times Z₂ represented by the matrices {1}⁽³⁾, \varphi ₁, \varphi ₂, \varphi ₃. The effective neutrino mixing matrix U with s₁₃ = 0 is always invariant under the whole Z₂ \times Z₂ represented in two ways, by the matrices {1}⁽³⁾, \varphi ₁, \varphi ₂, \varphi ₃ and {1}⁽³⁾, \mu ₁, \mu ₂, \mu ₃.

PACS numbers: 12.15.Ff, 14.60.Pq, 12.15.Hh