On a particular class of m-idempotent hyperrings, the relation ξm* is the smallest strongly regular equivalence such that the related quotient ring is commutative. Thus, on such hyperrings, ξm* is a new representation for the α*-relation. In this paper, the ξm-parts on hyperrings are defined and compared with complete parts, α-parts, and m-complete parts, as generalizations of complete parts in hyperrings. It is also shown how the ξm-parts help us to study the transitivity property of the ξm-relation. Finally, ξm-complete hyperrings are introduced and studied, stressing on the fact that they can be characterized by ξm-parts. The symmetry plays a fundamental role in this study, since the protagonist is an equivalence relation, defined using also the symmetrical group of permutations of order n.