  Neutrino oscillation data indicate that $\theta_{23}$ is close to $\pi/4$ and
$\theta_{13}$ is very small. A simple $\mu\leftrightarrow\tau$ exchange
symmetry of the neutrino mass matrix predicts $\theta_{23}=-\pi/4$ and
$\theta_{13}=0$. Since the experimental measurements differ from these
predictions, this symmetry is obviously broken. This breaking is given by two
parameters: $\varepsilon_1$ parametrizing the inequality bewteen $12$ and $13$
elements and $\varepsilon_2$ parametrizing the inequality bewteen $22$ and $33$
elements. We show that the magnitude of $\theta_{13}$ is essentially controlled
by $\varepsilon_1$ whereas the deviation of $\theta_{23}$ from maximality is
controlled by $\varepsilon_2$. The measured value of $\theta_{13}$ requires
$\mu\leftrightarrow\tau$ symmetry to be badly broken for both normal hierarchy
and inverted hierarchy, though the level of breaking depends sensitively on the
hierarchy. In this paper we obtain constraints on the parameters of neutrino
mass matrix, including the symmetry breaking parameters, using the precision
oscillation data. We find that this precision data constrains all elements of
neutrino mass matrix to be in very narrow ranges. We also consider
$\mu\leftrightarrow -\tau$ exchange symmetry in the case of inverted hierarchy
and find that it provides an explanation of neutrino mixing angles with some
fine-tuning.
