This paper studies the degree to which a bivariate copula fails to be symmetric under coordinate permutation, a property known as non-exchangeability. Working within an axiomatic framework that quantifies this asymmetry through a family of $L^p$-based measures, we establish sharp bounds linking non-exchangeability to classical dependence and concordance measures, prove exact scaling laws under convex mixing that enable explicit construction of copulas with any prescribed degree of asymmetry, and characterise the class of maximally non-exchangeable copulas together with the feasible range of asymmetry--concordance pairs. On the inferential side, we propose a nonparametric permutation test for exchangeability with exact finite-sample error control and consistency against all asymmetric alternatives, validated by Monte Carlo simulation and illustrated on a real data set.