Robust Covariance of Overlapping Sums of Independent Variables

Let $X$, $Y$, $Z$ be independent random variables with $\operatorname{Var}(X) = 1$, $\operatorname{Var}(Y) = 2$, and $\operatorname{Var}(Z) = 3$. Define $U = X + Y$ and $V = Y + Z$. Compute $\operatorname{Cov}(U, V)$ and $\operatorname{Corr}(U, V)$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.