  We carry out a phenomenological and analytical study of the texture
structures for the Majorana neutrino mass matrix characterized by one single
equality between two independent matrix elements, with vanishing non-physical
phases whose role in the definition we clarify. We find that fourteen textures
are viable in both types of hierarchy, whereas the remaining fifteenth one is
viable only in normal hierarchy. We also present symmetry realizations for some
patterns by using Abelian flavor symmetries within type-I, type-II and mixed
type-(I+II) seesaw scenarios.
