  For a given set of random variables $X_1,\ldots,X_d$ we seek as large a
family as possible of random variables $Y_1,\ldots,Y_d$ such that the marginal
laws and the laws of the sums match: $Y_i\,{\buildrel d \over =}\,X_i$ and
$\sum_iY_i\,{\buildrel d \over =}\,\sum_iX_i$. Under the assumption that
$X_1,\ldots,X_d$ are independent and belong to any of the Meixner classes, we
give a full characterisation of the random variables $Y_1,\ldots,Y_d$ and
propose a practical construction by means of a finite mean square expansion.
When $X_1,\ldots,X_d$ are identically distributed but not necessarily
independent, using a symmetry-balancing approach we provide a universal
construction with sufficient symmetry to satisfy the more stringent requirement
that, for any symmetric function $g$, $g(Y)\,{\buildrel d \over =}\,g(X)$.
