  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 $\sim$ meV range and (ii)
testable correlations among the various mixing angles, including a nonzero
$\theta_{13}$ angle with its exact value correlated with the values of the
atmospheric angle $\theta_{23}$ and the $CP$ phase $\delta$. 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.
