  The requirement of the mu-tau symmetry in the neutrino sector that yields the
maximal atmospheric neutrino mixing is shown to yield either sin(\theta_{13})=0
(referred to as C1)) or sin(\theta_{12})=0 (referred to as C2)), where
\theta_{12(13)} stands for the solar (reactor) neutrino mixing angle. We study
general properties possessed by approximately mu-tau symmetric textures. It is
argued that the tiny mu-tau symmetry breaking generally leads to
cos(2\theta_{23}) \simsin(\theta_{13}) for C1) and cos(2\theta_{23}) \sim
\Delta m^2_\odot/\Delta m^2_{atm}(\equiv R) for C2), which indicates that the
smallness of cos(2\theta_{23}) is a good measure of the mu-tau symmetry
breaking, where \Delta m^2_{atm} (\Delta m^2_\odot) stands for the square mass
differences of atmospheric (solar) neutrinos. We further find that the relation
R \sim sin^2(\theta_{13}) arises from contributions of O(sin^2(\theta_{13})) in
the estimation of the neutrino masses (m_{1,2,3}) for C1), and that possible
forms of textures are strongly restricted to realize sin^2(2\theta_{12})=O(1)
for C2). To satisfy R \sim sin^2(\theta_{13}) for C1), neutrinos exhibit the
inverted mass hierarchy, or the quasi degenerate mass pattern with | m_{1,2,3}|
\sim O(\sqrt{\Delta m^2_{atm}}), and, to realize sin^2(2\theta_{12})=O(1) for
C2), there should be an additional small parameter \eta whose size is
comparable to that of the mu-tau symmetry breaking parameter \epsilon, giving
tan(2\theta_{12}) \sim \epsilon/\eta with \eta \sim \epsilon to be compatible
with the observed large mixing.
