The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity and consequently its vertices can be repositioned by folding (changing the dihedral angle) at edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.
Rigid and kinematic cuboctahedra
When interpreted as a framework of rigid flat faces, connected along the edges by hinges, the cuboctahedron is a rigid structure, as are all convex polyhedra, by Cauchy's theorem. However, when the faces are removed, leaving only rigid edges connected by flexible joints at the vertices, the result is not a rigid system (unlike polyhedra whose faces are all triangles, to which Cauchy's theorem applies despite the missing faces).
Adding a central vertex, connected by rigid edges to all the other vertices, subdivides the cuboctahedron into square pyramids and tetrahedra, meeting at the central vertex. Unlike the cuboctahedron itself, the resulting system of edges and joints is rigid, and forms part of the infinite octet truss structure.
Cyclical cuboctahedron transformations
The cuboctahedron can be transformed cyclically through four polyhedra, repeating the cycle endlessly. Topologically the transformation follows a Möbius loop: it is an orientable double cover of the octahedron.
In their spatial relationships the cuboctahedron, icosahedron, Jessen's icosahedron, and octahedron nest like Russian dolls and are related by a helical contraction. The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles. The 12 vertices of the cuboctahedron spiral inward (toward the center) and move closer together until they reach the points where they form a regular icosahedron; they move slightly closer together until they form a Jessen's icosahedron; and they continue to spiral toward each other until they coincide in pairs as the 6 vertices of the octahedron.
The general cuboctahedron transformation can be parameterized along a continuum of special-case transformations with two limit-cases: one in which the edges of the cuboctahedron are rigid, and one in which they are elastic.
Rigid-edge transformation
The rigid-edge cuboctahedron transformation symmetrically transforms the cuboctahedron into a regular icosahedron, a Jessen's icosahedron, and a regular octahedron, in the sense that the polyhedron's vertices take on the vertex positions of those polyhedra successively.
The cuboctahedron does not actually become those other polyhedra, and they cannot transform into each other (if they have rigid edges), because unlike the cuboctahedron they do have structural rigidity as a consequence of having only triangular faces.
What the cuboctahedron with rigid edges actually can transform into (and through) is a regular icosahedron from which 6 edges are missing (a pseudoicosahedron), a Jessen's icosahedron in which the 6 reflex edges are missing or elastic, and a double cover of the octahedron that has two coincident rigid edges connecting each pair of vertices (formed by making pairs of cuboctahedron vertices coincide).
Elastic-edge transformation
There is a tensegrity polyhedron which embodies and enforces the closely related elastic-edge cuboctahedron transformation. The tensegrity icosahedron has a dynamic structural rigidity called infinitesimal mobility and can only be deformed into symmetrical polyhedra along that spectrum from cuboctahedron to octahedron. It is called the tensegrity icosahedron because its median stable form is the Jessen's icosahedron.