  MoTe$_2$ has recently attracted much attention due to the observation of
pressure-induced superconductivity, exotic topological phase transitions, and
nonlinear quantum effects. However, there has been debate on the intriguing
structural phase transitions among various observed phases of MoTe$_2$, and
their connection to the underlying topological electronic properties. In this
work, by means of density-functional theory (DFT+U) calculations, we
investigate the structural phase transition between the polar T$_d$ and
nonpolar 1T$'$ phases of MoTe$_2$ in reference to a hypothetical high-symmetry
T$_0$ phase that exhibits higher-order topological features. In the T$_d$ phase
we obtain a total of 12 Weyl points, which can be created/annihilated,
dynamically manipulated, and switched by tuning a polar phonon mode. We also
report the existence of a tunable nonlinear Hall effect in T$_d$-MoTe$_2$, and
propose the use of this effect as a probe for the detection of polarity
orientation in polar (semi)metals. By studying the role of dimensionality, we
identify a configuration in which a nonlinear surface response current emerges.
The potential technological applications of the tunable Weyl phase and the
nonlinear Hall effect are discussed.
