For quasiperiodic flows of Koch type, we exploit an algebraic rigidity of an equivalence relation on flows, called projective conjugacy, to algebraically characterize the deviations from completeness of an absolute invariant of projective conjugacy, called the multiplier group, which describes the generalized symmetries of the flow. We then describe three ways by which two quasiperiodic flows with the same Koch field are projectively conjugate when their multiplier groups are identical. The first way involves a quantity introduced here, called the G-paragon class number of the multiplier group. The second involves the generalized Bowen-Franks groups and the class number of an order. The third involves conjugacy of the actions of the multiplier group by commuting toral automorphisms, for which one of these actions is irreducible, and a condition introduced here, called PCF, on the common real eigenvectors of the irreducible action. Additionally, we describe two ways by which similiar actions of the multiplier group can fail to be conjugate.