A consensus index method comprises a consensus method and a consensus index that are defined on a common set of objects (e.g. classifications). For each profile of objects, the consensus method returns a consensus object representing information or structure shared among profile objects, while the consensus index returns a quantitative measure of agreement among profile objects. Since the relationship between consensus method and consensus index is poorly understood, we propose simple axioms prescribing it in the most general terms. Many taxonomic consensus index methods violate these axioms because their consensus indices measure consensus object invariants rather than profile agreement. We propose paradigms to obtain consensus index methods that measure agreement and satisfy the axioms. These paradigms salvage concepts underlying consensus index methods violating the axioms.