  We report a time-reversal method based on the Richards-Wolf vectorial
diffraction theory to generate transversely oriented optical M\"obius strips
that wander around an axis perpendicular to the beam propagation direction. A
number of sets of dipole antennae are purposefully positioned on a prescribed
trajectory in the y = 0 plane and the radiation fields are collected by one
high-NA objective lens. By sending the complex conjugate of the radiation
fields in a time-reversed manner, the focal fields are calculated and the
optical polarization topology on the trajectory can be tailored to form
prescribed M\"obius strips. The method can be extended to construct various
polarization topologies on three-dimensional trajectories in the focal region.
The ability to control optical polarization topologies may find applications in
nanofabrication, quantum communications, and light-matter interactions.
