Although tetrameric hemoglobin has been studied extensively as a prototype for understanding mechanisms of allosteric regulation, the functional and structural properties of its eight intermediate ligation forms have remained elusive. Recent experiments on the energetics of cooperativity of these intermediates, along with assignments of their quaternary structures, have revealed that the allosteric mechanism is controlled by a previously unrecognized symmetry feature: quaternary switching from form T to form R occurs whenever heme-site binding creates a tetramer with at least one ligated subunit on each dimeric half-molecule. This "symmetry rule" translates the configurational isomers of heme-site ligation into six observed switchpoints of quaternary transition. Cooperativity arises from both "concerted" quaternary switching and "sequential" modulation of binding within each quaternary form, T and R. Binding affinity is regulated through a hierarchical code of tertiary-quaternary coupling that includes the classical allosteric models as limiting cases.