We discuss the connection between the origin of neutrino masses and their mixings, which arises in a class of scenarios with radiatively induced neutrino masses. In these scenarios, the neutrino mass matrix acquires textures with two entries close to zero in the basis where the charged-lepton mass matrix is diagonal. This results in specific constraints on the neutrino mixing parameters, which leads to the prediction of (i) a normal ordering of neutrino masses with the lightest neutrino mass in the ∼meV range and (ii) testable correlations among the various mixing angles, including a nonzero θ13 angle with its exact value correlated with the values of the atmospheric angle θ23 and the CP phase δ. We quantify the impact of deviations from exact zeroes in the mass matrix texture and connect it to the amount of hierarchy among Yukawa couplings. These scenarios of radiative neutrino mass generation also give rise to new short-range contributions to neutrinoless double beta decay, which dominate over the usual light-neutrino exchange contribution. As a result, this class of models can have a sizable neutrinoless double beta decay rate, in the range of upcoming experiments despite the normal mass ordering of neutrinos. © 2014 American Physical Society.