Quasi-symmetry is symmetry of the interaction component and geometric marginal homogeneity is symmetry of the independence component of a square probability table, so skewness decomposes as a sum of two orthogonal measures.
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Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables
Quasi-symmetry is symmetry of the interaction component and geometric marginal homogeneity is symmetry of the independence component of a square probability table, so skewness decomposes as a sum of two orthogonal measures.