  We determine a set of necessary conditions on a partition-indexed family of
complex numbers to be the "highest coefficients" of a positive and symmetric
multi-faced universal product; i.e. the product associated with a multi-faced
version of noncommutative stochastic independence, such as bifreeness. The
highest coefficients of a universal product are the weights of the
moment-cumulant relation for its associated independence. We show that these
conditions are almost sufficient, in the sense that whenever the conditions are
satisfied, one can associate a (automatically unique) symmetric universal
product with the prescribed highest coefficients. Furthermore, we give a quite
explicit description of such families of coefficients, thereby producing a list
of candidates that must contain all positive symmetric universal products. We
discover in this way four (three up to trivial face-swapping) previously
unknown moment-cumulant relations that give rise to symmetric universal
products; to decide whether they are positive, and thus give rise to
independences which can be used in an operator algebraic framework, remains an
open problem.
