  Motivated by the recent experimental realization of twisted transition metal
dichalcogenide bilayers, we study a simplified model driven by different forms
of monochromatic light. As a concrete and representative example we use
parameters that correspond to a twisted MoTe$_2$ homobilayer. First, we
consider irradiation with circularly polarized light in free space and
demonstrate that the corresponding Floquet Hamiltonian takes the same form as
the static Hamiltonian, only with a constant overall shift in quasi-energy.
This is in stark contrast to twisted bilayer graphene, where new terms are
typically generated under an analagous drive. Longitudinal light, on the other
hand, which can be generated from the transverse magnetic mode in a waveguide,
has a much more dramatic effect--it renormalizes the tunneling strength between
the layers, which effectively permits the tuning of the twist angle {\em
in-situ}. We find that, by varying the frequency and amplitude of the drive,
one can induce a topological transition that cannot be obtained with the
traditional form of the Floquet drive in free space. Furthermore, we find that
strong drives can have a profound effect on the layer pseudospin texture of the
twisted system, which coincides with multiple simultaneous band gap closings in
the infinite-frequency limit. Surprisingly, these bandgap closings are not
associated with topological transitions. For high but finite drive frequencies
near $0.7$eV, the infinite-frequency band crossings become band gap minima of
the order of $10^{-6}$ eV or smaller.
