  We study the implications of the existence of two equalities between the
elements or cofactors of the neutrino mass matrix. There are sixty five
structures of this type for each case. Phenomenological implications for
unknown parameters like the effective Majorana mass of the electron neutrino
and CP-violating phases are examined for the viable cases. To illustrate how
such forms of the neutrino mass matrices may be realized, we also present a
simple $A_4$ model for one of the classes in each case.
