The mitochondrial chaperonin, mtHsp60, promotes the folding of newly imported and transiently misfolded proteins in the mitochondrial matrix, assisted by its co-chaperone mtHsp10. Despite its essential role in mitochondrial proteostasis, structural insights into how this chaperonin binds to clients and progresses through its ATP-dependent reaction cycle are not clear. Here, we determined cryo-electron microscopy (cryo-EM) structures of a hyperstable disease-associated mtHsp60 mutant, V72I, at three stages in this cycle. Unexpectedly, client density is identified in all states, revealing interactions with mtHsp60's apical domains and C-termini that coordinate client positioning in the folding chamber. We further identify a striking asymmetric arrangement of the apical domains in the ATP state, in which an alternating up/down configuration positions interaction surfaces for simultaneous recruitment of mtHsp10 and client retention. Client is then fully encapsulated in mtHsp60/mtHsp10, revealing prominent contacts at two discrete sites that potentially support maturation. These results identify a new role for the apical domains in coordinating client capture and progression through the cycle, and suggest a conserved mechanism of group I chaperonin function.