  Hints for sizable $\sin^2 \theta_{13}$ have been reported in earlier global
neutrino oscillation data analyses as well as will be reported in this work,
and quite recently by the Double Chooz experiment. However, as we enter the era
of precision neutrino oscillation experiments, terms linear in
$\sin\theta_{13}$ will no longer be negligible, and its sign would affect the
extraction of other oscillation parameters. The sign of $\sin\theta_{13}$ also
plays a crucial role in the determination of the CP-violating phase. In this
work we show that by adopting an alternative parametrization for the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix, one already has a chance
to infer the sign of each mixing angle in the conventional parametrization
using existing global neutrino data. A weak preference for negative $\sin
\theta_{13}$ is found. In particular, the solar data suggest that
$\sin\theta_{13} > 0$ while all other data the opposite. This leads to the
speculation on whether the Mikheyev-Smirnov-Wolfenstein (MSW) effect is
responsible. In this work we found that in the new mixing matrix
parametrization, the 68% CL constraints on the three mixing angles are
comparable to those estimated in the conventional parametrization adopted in
the literature. Owing to the strong correlations among the three mixing angles
in the new parametrization, the advantages of doing the global neutrino
oscillation analysis using data from past, current, and near future neutrino
oscillation experiments shall become manifest.
