  We establish a classification of cubic minimal cones in case of the so-called
radial eigencubics. Our principal result states that any radial eigencubic is
either a member of the infinite family of eigencubics of Clifford type, or
belongs to one of 18 exceptional families. We prove that at least 12 of the 18
families are non-empty and study their algebraic structure. We also establish
that any radial eigencubic satisfies the trace identity $\det \mathrm{Hess}^3
(f)=\alpha f$ for the Hessian matrix of $f$, where $\alpha\in \R{}$. Another
result of the paper is a correspondence between radial eigencubics and
isoparametric hypersurfaces with four principal curvatures.
