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Non-exchangeable dependence structures exist in the real world, and we are interested in how to identify the existence of non-exchangeability in the joint distributional tails and how to quantify the degree of such tail non-exchangeability. The results obtained and the approaches proposed benefit bivariate dependence modeling when dependence patterns in the tails are particularly important, as in the fields of quantitative finance, quantitative risk management, and econometrics. We focus on the bivariate case, and propose to use conditional expectations as the basis quantities. Then, for random variables X and Y, the departure between tail behavior of E[XY/> t] and E[Y/X > t], or E[X/Y = t] and E[Y/X = t], when t is large, becomes sensible in detecting tail non-exchangeability. We use a bivariate copula to model the dependence between X and Y. Various devices of generating non-exchangeable copulas as well as three major tail behaviors for univariate margins are studied, in order to understand the interaction between the departure of those conditional expectations and the non-exchangeable dependence together with various types of margins. Based on the probabilistic properties of the tail non-exchangeability structures, we develop graphical approaches and statistical tests for analyzing dataset that may have non-exchangeability in the joint tail. Simulation study and empirical study are then conducted to demonstrate the usefulness of the proposed approaches.