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 $\cos 2{\theta }_{23}\sim \sin {\theta }_{13}$ for C1) and $\cos 2{\theta }_{23}\sim \Delta m_{\odot }^2/\Delta m_{\mathrm{atm}}^2(\equiv R)$ for C2), which indicates that the smallness of $\cos 2{\theta }_{23}$ is a good measure of the $\mu -\tau$ symmetry breaking, where $\Delta m_{\mathrm{atm}}^2$ ($\Delta m_{\odot }^2$) stands for the square mass difference of atmospheric (solar) neutrinos. We further find that the relation $R\sim {sin}^2{\theta }_{13}$ arises from contributions of $\mathcal{O}({sin}^2{\theta }_{13})$ in the estimation of the neutrino masses ($m_{1,2,3}$) for C1), and that possible forms of textures are strongly restricted to realize ${sin}^{2}2{\theta }_{12}=\mathcal{O}(1)$ 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 $|m_{1,2,3}|\sim \mathcal{O}(\sqrt{\Delta m_{\mathrm{atm}}^2})$, and, to realize ${sin}^{2}2{\theta }_{12}=\mathcal{O}(1)$ 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.

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