mHsp60-mHsp10 assists the folding of mitochondrial matrix proteins without the negative ATP binding inter-ring cooperativity of GroEL-GroES. Here we report the crystal structure of an ATP (ADP:BeF3-bound) ground-state mimic double-ring mHsp6014-(mHsp107)2 football complex, and the cryo-EM structures of the ADP-bound successor mHsp6014-(mHsp107)2 complex, and a single-ring mHsp607-mHsp107 half-football. The structures explain the nucleotide dependence of mHsp60 ring formation, and reveal an inter-ring nucleotide symmetry consistent with the absence of negative cooperativity. In the ground-state a two-fold symmetric H-bond and a salt bridge stitch the double-rings together, whereas only the H-bond remains as the equatorial gap increases in an ADP football poised to split into half-footballs. Refolding assays demonstrate obligate single- and double-ring mHsp60 variants are active, and complementation analysis in bacteria shows the single-ring variant is as efficient as wild-type mHsp60. Our work provides a structural basis for active single- and double-ring complexes coexisting in the mHsp60-mHsp10 chaperonin reaction cycle.