Based on a new approach for measuring the comovements between stock market returns, we provide a nonparametric test for asymmetric comovements in the sense that stock market downturns will lead to stronger comovements than market upturns. The test is used to detect whether asymmetric comovements exist in international stock markets. We find the following empirical facts. First, asymmetric comovements exist between the United States (U.S.) stock market and the stock markets for Canada, France, Germany, and the United Kingdom (U.K.), but the data are unable to reject the null hypothesis of the symmetric comovements between the U.S. and Japanese stock markets. Second, either a larger negative drop or a positive increase in stock prices leads to stronger comovements of stock market returns, indicating that comovements in the data are different from comovements implied by a bivariate symmetric distribution, which implies that comovements tend to zero as the market returns become more positive or more negative.