  We report curling self-propulsion in aqueous emulsions of common mesogenic
compounds. Nematic liquid crystal droplets self-propel in a surfactant solution
with concentrations above the critical micelle concentration while undergoing
micellar solubilization. We analyzed trajectories both in a Hele-Shaw geometry
and in a 3D setup at variable buoyancy. The coupling between the nematic
director field and the convective flow inside the droplet leads to a second
symmetry breaking which gives rise to curling motion in 2D. This is
demonstrated through a reversible transition to non-helical persistent swimming
by heating to the isotropic phase. Furthermore, auto-chemotaxis can
spontaneously break the inversion symmetry, leading to helical trajectories.
