  Type-II Weyl fermion nodes, located at the touching points between electron
and hole pockets, have been recently predicted to occur in distorted octahedral
($T'$) transition metal dichalcogenide semimetals, contingent upon the
condition that the layered crystal has the noncentrosymmetric orthorhombic
($T'_{or}$) stacking. Here, we report on the emergence of two shear Raman bands
activated by inversion symmetry breaking in $T'$-MoTe$_2$ due to sample
cooling. Polarization and crystal orientation resolved measurements further
point to a phase transition from the monoclinic ($T'_{mo}$) structure to the
desired $T'_{or}$ lattice. These results provide spectroscopic evidence that
low-temperature $T'$-MoTe$_2$ is suitable for probing type-II Weyl physics.
