Breaking down the sum of two random variables

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Is there a pair of continuous and independent random variables $X$ and $Y$, such that you can infer the values of $X$ and $Y$ given their sum ? A discrete example would be $X$ uniform on $\{0,2\}$ and $Y$ uniform on $\{3,4\}$, so the sum $X+Y$ (equal to $3,4,5$ or $6$) entirely determines the values of $X$ and $Y$.