We study the phenomenological implications of the recent neutrino data for class B of two texture zeros and two vanishing cofactors for Majorana neutrinos in the flavor basis. We find that the classes B1 ( B2 ) of two texture zeros and the classes B5 ( B6 ) of two vanishing cofactors have similar predictions for neutrino oscillation parameters for the same mass hierarchy. Similar predictions for the classes B3 ( B4 ) of two texture zeros and classes B3 ( B4 ) of two vanishing cofactors are expected. However, a preference for a shift in the quadrant of the Dirac-type CP-violating phase ( δ ) in contrast to the earlier analysis has been predicted for a relatively large value of the reactor neutrino mixing angle ( θ13 ) for class B of two text...