  We study a texture of neutrino mass matrix characterized by two constraints
consisting of one equality and another antiequality between two elements
corresponding to two pairs of the matrix entries. Amidst such textures, we
limit our study to three patterns which were realizable assuming an
$S_4$-symmetry within type II-seesaw scenario. Three such cases were found and
studied: I ($M_{\n 22}=-M_{\n 33}$ \& $M_{\n 11}=+M_{\n 23}$), II ($M_{\n
11}=-M_{\n 33}$ \& $M_{\n 22}=+M_{\n 13}$) and III ($M_{\n 11}=-M_{\n 22}$ \&
$M_{\n 33}=+M_{\n 12}$). We specify the role of unphysical phases in the
definition of the textures under study which were tested against experimental
constraints, and were found to accommodate data with both hierarchies allowed.
However, switching off the unphysical phases allows only for inverted
hierarchy, except for the texture III which allows also, albeit for a very
narrow parameter space region, for normal ordering. We stress that the
different phenomenologies when including/excluding unphysical phases stem from
the different definitions of the texture one has to adopt in order to make it
insensitive to unphysical phases, rather than to any `absent' physical effects
of unphysical phases. We present a complete phenomenological analysis of these
three textures and justify analytically the resulting correlations. We detail
the effect of the unphysical phases in diluting/deforming several correlations,
which otherwise would have been ``clear". Finally, we give theoretical
realizations within seesaw type II scenarios for such textures.
