The theory of marginal convection in a uniformly rotating, self-gravitating, fluid sphere, of uniform density and containing a uniform distribution of heat sources, is developed to embrace modes of convection which are asymmetric with respect to the axis of rotation. It is shown that these modes are the most unstable, except for the smallest Taylor numbers, T (a measure of the rotation rate); i.e. for any T and $\omega $ (Prandtl number), the lowest Rayleigh number (a measure of the temperature gradients in the sphere) is associated with an asymmetric motion. This is demonstrated both by an expansion method suitable for small T, and by asymptotic theory for T $\rightarrow \infty $. For large T, the eigenmode most easily excited is small in amplitude everywhere except near a cylindrical surface, of radius about half that of the sphere, and coaxial with the diameter parallel to the angular velocity vector.