  Spin-orbit coupling (SOC) in solids normally originates from the electron
motion in the electric field of the crystal. It is key to understanding a
variety of spin-transport and topological phenomena, such as Majorana fermions
and recently discovered topological insulators. Implementing and controlling
spin-orbit coupling is thus highly desirable and could open untapped
opportunities for the exploration of unique quantum physics. Here, we show that
dipole-dipole interactions can produce an effective SOC in two-dimensional
ultracold polar molecule gases. This SOC generates chiral excitations with a
non-trivial Berry phase $2\pi$. These excitations, which we call
\emph{chirons}, resemble low-energy quasiparticles in bilayer graphene and
emerge regardless of the quantum statistics and for arbitrary ratios of kinetic
to interaction energies. Chirons manifest themselves in the dynamics of the
spin density profile, spin currents, and spin coherences, even for molecules
pinned in a deep optical lattice and should be observable in current
experiments.
