  It is shown that the neutrino mass matrices in the flavour basis yielding a
vanishing $U_{e3}$ are characterized by invariance under a class of $Z_2$
symmetries. A specific $Z_2$ in this class also leads to a maximal atmospheric
mixing angle $\theta_{23}$. The breaking of that $Z_2$ can be parameterized by
two dimensionless quantities, $\e$ and $\e'$; the effects of $\e, \e' \neq 0$
are studied perturbatively and numerically. The induced value of $\ue3$
strongly depends on the neutrino mass hierarchy. We find that $\ue3$ is less
than 0.07 for a normal mass hierarchy, even when $\e, \e' \sim 30 %$. For an
inverted mass hierarchy $\ue3$ tends to be around 0.1 but can be as large as
0.17. In the case of quasi-degenerate neutrinos, $\ue3$ could be close to its
experimental upper bound 0.2. In contrast, $| \cos{2\theta_{23}} |$ can always
reach its experimental upper bound 0.28. We propose a specific model, based on
electroweak radiative corrections in the MSSM, for $\e$ and $\e'$. In that
model, both $\ue3$ and $| \cos{2 \theta_{23}} |$, could be close to their
respective experimental upper bounds if neutrinos are quasi-degenerate.
