[en] We study the implications of the simultaneous existence of a texture zero and a vanishing minor in the neutrino mass matrix. There are 36 possible texture structures of this type, 21 of which reduce to two texture zero cases which have, already, been extensively studied. Of the remaining 15 textures only six are allowed by the current data. We examine the phenomenological implications of the allowed texture structures for Majorana type CP-violating phases, 1-3 mixing angle, and Dirac type CP-violating phase. All these possible textures can be generated through the seesaw mechanism and realized in the framework of discrete Abelian flavor symmetry. We present the symmetry realization of these texture structures.