Sign reversal under reversal of one ordinal ordering ($\gamma \mapsto -\gamma$)

Reverse the order of exactly one of two ordinal categorical variables. Every concordant pair becomes a discordant pair and conversely, while pairs tied on either variable remain tied. Therefore the numerator $C-D$ changes sign and the denominator $C+D$ does not: Goodman and Kruskal's gamma changes sign whenever defined. The same argument applies to Somers' D, Kendall's tau-b and Stuart's tau-c, whose denominators depend on tie counts or table dimensions rather than the orientation. Reversing both orders preserves the sign. This explains why an association's numerical sign must be interpreted using the actual coding directions; it does not change the substantive paired observations. Measures of nominal association such as Cramér's V are invariant under either reversal.