Correlation of Sum and Difference

Let $X$ and $Y$ be i.i.d. random variables, each with variance $\sigma^2$. Define $S = X + Y$ and $D = X - Y$. What is the correlation $\rho(S, D)$? As a follow-up: if $X$ and $Y$ are i.i.d. normal, can you say something stronger than just their correlation?