We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G2 structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G2 structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff–Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.