Every copula $ C $ for a random vector $ {\bf X}=(X_1,\dots,X_d) $ with identically distributed coordinates determines a unique copula $ C_{:d} $ for its order statistic $ {\bf X}_{:d}=(X_{1:d},\dots,X_{d:d}) $. In the present paper we study the dependence structure of $ C_{:d} $ via Kendall's tau, denoted by $ κ$. As a general result, we show that $ κ[C_{:d}] $ is at least as large as $ κ[C] $. For the product copula $ Π$, which corresponds to the case of independent coordinates of $ {\bf X} $, we provide an explicit formula for $ κ[Π_{:d}] $ showing that the inequality between $ κ[Π] $ and $ κ[Π_{:d}] $ is strict. We also compute Kendall's tau for certain multivariate margins of $ Π_{:d} $ corresponding to the lower or upper coordinates of $ {\bf X}_{:d} $.