Abstract A new method for the α,β-itemized enumeration has been developed to count inositol derivatives, where each substitution position of a hexagonal skeleton accommodates an α-ligand or a β-ligand exclusively under the D6- or D6h-symmetry. The concept of hedralities of cycles (homohedral cycles, enantiohedral cycles, and hemihedral cycles) has been proposed to treat α,β-handedness for characterizing such exclusive α- or β-substitution. During the α,β-itemized enumeration, the concept of hedralities is found to cooperate with the concept of sphericities of cycles (homospheric cycles, enantiospheric cycles, and hemispheric cyles), which was proposed to treat chirality in Fujita’s proligand method. The effect of hedralities is evaluated by developing the cycle index with α,β-handedness fittingness (CI-HF), which enables us to accomplish the α,β-itemized enumeration under D6. The cooperative effect of hedralities and sphericities can be evaluated by extending Fujita’s proligand method, where the cycle index with pairwise fittingness (CI-PF) is developed to support the α,β-itemized enumeration under D6h. The GAP codes for the enumeration using CI-HFs and CI-PFs are programmed on the basis of the GAP system. The enumeration method based on CI-HFs and CI-PFs is generalized to support the α,β-itemized enumeration of m-gonal derivatives under Dm- or Dmh-symmetry. The method for gross enumeration without α,β-itemization and without ligand discrimination is developed as a special case of the extended proligand method described above. General formulas are obtained to give gross numbers as integer sequences.