Riemannian cubics are curves in a Riemannian manifold M satisfying a variational condition. They arise in computer graphics and in trajectory planning problems for rigid body motion, where M is the group SO(3) of rotations of Euclidean three-space E-3. Riemannian cubics on a Lie group correspond to Lie quadratics in the Lie algebra. There are only a few cases where closed-form expressions are available for Lie quadratics. The present article is a qualitative analysis of null quadratics in so(3), focusing on long term dynamics and internal symmetries. Conclusions are drawn for asymptotics and symmetries of null cubics in SO(3). (C) 2003 American Institute of Physics.