  Motivated by the experimental observation of spin-charge separation in
one-dimensional interacting Fermi gases, we investigate these systems in the
presence of spin-orbit coupling and Rabi fields. We demonstrate that
spin-charge-separated modes evolve into helical collective modes due to the
special mixing of spin and charge induced by spin-orbit coupling and Rabi
fields. We obtain the phase diagram of chemical potential versus Rabi fields
for given spin-orbit coupling and interactions, and find several topological
quantum phase transitions of the Lifshitz type. We show that the velocities of
the collective modes are nonanalytic at the boundaries between different
phases. Lastly, we analyze the charge-charge, spin-charge and spin-spin
dynamical structure factors to show that the dispersions, spectral weights and
helicities of the collective modes can be experimentally extracted in systems
such as $^{6}{\rm Li}$, $^{40}{\rm K}$ and $^{173}{\rm Yb}$.
