The mature retrovirus capsid consists of a variably curved lattice of capsid protein (CA) hexamers and pentamers. High-resolution structures of the curved assembly, or in complex with host factors, have not been available. By devising cryo-EM methodologies for exceedingly flexible and pleomorphic assemblies, we have determined cryo-EM structures of apo-CA hexamers and in complex with cyclophilin A (CypA) at near-atomic resolutions. The CA hexamers are intrinsically curved, flexible and asymmetric, revealing the capsomere and not the previously touted dimer or trimer interfaces as the key contributor to capsid curvature. CypA recognizes specific geometries of the curved lattice, simultaneously interacting with three CA protomers from adjacent hexamers via two noncanonical interfaces, thus stabilizing the capsid. By determining multiple structures from various helical symmetries, we further revealed the essential plasticity of the CA molecule, which allows formation of continuously curved conical capsids and the mechanism of capsid pattern sensing by CypA.