Proof of Hoeffding's Covariance Identity
Solution 1:
It suffices to observe that the random variables $\mathbb 1_{\{X \le x\}}$ and $\mathbb 1_{\{X \ge x\}}$ are perfectly correlated (except on a set of measure 0). Specifically, their sum is almost surely 1. Since the same holds for the indicator for $Y$, it immediately follows that the covariance of $\mathbb 1_{\{X \le x\}}$ and $\mathbb 1_{\{Y \le y\}}$ will be equal to the covariance of $\mathbb 1_{\{X \ge x\}}$ and $\mathbb 1_{\{Y \ge y\}}$.