  For every proper convex cone $K \subset \mathbb R^3$ there exists a unique
complete hyperbolic affine 2-sphere with mean curvature $-1$ which is
asymptotic to the boundary of the cone. Two cones are associated if the
corresponding affine spheres can be mapped to each other by an
orientation-preserving isometry. This equivalence relation is generated by the
groups $SL(3,\mathbb R)$ and $S^1$, where the former acts by linear
transformations of the ambient space, and the latter by multiplication of the
cubic holomorphic differential of the affine sphere by unimodular complex
constants. The action of $S^1$ generalizes conic duality, which acts by
multiplication of the cubic differential by $-1$. We call a cone
self-associated if it is linearly isomorphic to all its associated cones, in
which case the action of $S^1$ induces (nonlinear) isometries of the
corresponding affine sphere. We give a complete classification of the
self-associated cones and compute isothermal parametrizations of the
corresponding affine spheres. The solutions can be expressed in terms of
degenerate Painlev\'e III transcendents. The boundaries of generic
self-associated cones can be represented as conic hulls of vector-valued
solutions of a certain third-order linear ordinary differential equation with
periodic coefficients, but there exist also cones with polyhedral boundary
parts.
