  I attempt to quantify how far from maximal one should expect the atmospheric
mixing angle to be given a neutrino mass-matrix that leads, at zeroth order, to
a nu_3 mass-eigenstate that is 0% nu_e, 50% nu_mu, and 50% nu_tau. This is done
by assuming that the solar mass-squared difference is induced by an
"anarchical" first order perturbation, an approach than can naturally lead to
experimentally allowed values for all oscillation parameters. In particular,
both |cos 2theta_atm| (the measure for the deviation of atmospheric mixing from
maximal) and |U_e3| are of order sqrt(Delta m^2_sol/Delta m^2_atm) in the case
of a normal neutrino mass-hierarchy, or of order Delta m^2_sol/Delta m^2_atm in
the case of an inverted one. Hence, if any of the textures analyzed here has
anything to do with reality, next-generation neutrino experiments can see a
nonzero cos 2theta_atm in the case of a normal mass-hierarchy, while in the
case of an inverted mass-hierarchy only neutrino factories should be able to
see a deviation of sin^2 2theta_atm from 1.
