On orthogonality of (X+Y)(X+Y) and X/(X+Y)X/(X+Y) rather than independence

If XX and YY are independent and if X+YX+Y and X/(X+Y)X/(X+Y) are independent random variables, then XX and YY must have gamma distributions. To confirm that lack of correlation between XX and X/(X+Y)X/(X+Y) does not characterize the gamma distribution, a large class of distributions are identified for which cov[X,X/(X+Y)]=0. A related question in the context of matrix-variate distributions is addressed.