Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $Ω$ determined by all possible values of Kendall's $τ$ and Spearman's $ρ$. In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is only sharp on a countable set with sole accumulation point $(-1,-1)$, give a simple analytic characterization of $Ω$ in terms of a continuous, strictly increasing piecewise concave function, and show that $Ω$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in Ω$ there are mutually completely dependent random variables whose $τ$ and $ρ$ values coincide with $x$ and $y$ respectively.