The development of new types of bonds and linkages that can reversibly tune the geometry and structural features of molecules is an elusive goal in chemistry. Herein, we report the use of catenated DNA structures as nanolinkages that can reversibly switch their angle and form different kinds of polygonal nanostructures. We designed a reconfigurable catenane that can self-assemble into a triangular or hexagonal structure upon addition of programmable DNA strands that function via toehold strand-displacement. The nanomechanical and structural features of these catenated nanojoints can be applied for the construction of dynamic systems such as molecular motors with switchable functionalities.