An accurate assessment of tail inequalities and tail asymmetries of financial returns is key for risk management and portfolio allocation. We propose a new test procedure for detecting the full extent of such structural differences in the dependence of bivariate extreme returns. We decompose the testing problem into piecewise multiple comparisons of Cramér-von Mises distances of tail copulas. In this way, tail regions that cause differences in extreme dependence can be located and consequently be targeted by financial strategies. We derive the asymptotic properties of the test and provide a bootstrap approximation for finite samples. Moreover, we account for the multiplicity of the piecewise tail copula comparisons by adjusting individual p-values according to multiple testing techniques. Monte Carlo simulations demonstrate the test’s superior finite-sample properties for common financial tail risk models, both in the i.i.d. and the sequentially dependent case. During the last 90 years in US stock markets, our test detects up to 20% more tail asymmetries than competing tests. This can be attributed to the presence of non-standard ta ... mehril dependence structures. We also find evidence for diminishing tail asymmetries during every major financial crisis – except for the 2007-09 crisis – reflecting a risk-return trade-off for extreme returns.