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We study a texture of neutrino mass matrix characterized by two constraintsconsisting of one equality and another antiequality between two elementscorresponding to two pairs of the matrix entries. Amidst such textures, welimit our study to three patterns which were realizable assuming an$S_4$-symmetry within type II-seesaw scenario. Three such cases were found andstudied: I ($M_{\n 22}=-M_{\n 33}$ \& $M_{\n 11}=+M_{\n 23}$), II ($M_{\n11}=-M_{\n 33}$ \& $M_{\n 22}=+M_{\n 13}$) and III ($M_{\n 11}=-M_{\n 22}$ \&$M_{\n 33}=+M_{\n 12}$). We specify the role of unphysical phases in thedefinition of the textures under study which were tested against experimentalconstraints, and were found to accommodate data with both hierarchies allowed.However, switching off the unphysical phases allows only for invertedhierarchy, except for the texture III which allows also, albeit for a verynarrow parameter space region, for normal ordering. We stress that thedifferent phenomenologies when including/excluding unphysical phases stem fromthe different definitions of the texture one has to adopt in order to make itinsensitive to unphysical phases, rather than to any `absent' physical effectsof unphysical phases. We present a complete phenomenological analysis of thesethree textures and justify analytically the resulting correlations. We detailthe effect of the unphysical phases in diluting/deforming several correlations,which otherwise would have been ``clear". Finally, we give theoreticalrealizations within seesaw type II scenarios for such textures.

Textures of Neutrino Mass Matrix from $S_4$-flavor Symmetry