  The reduced peripheral system was introduced by Milnor in the fifties for the
study of links up to link-homotopy, i.e. up to isotopies and crossing changes
within each link component. However, for four or more components, this
invariant does not yield a complete link-homotopy invariant. This paper
provides two characterizations of links having the same reduced peripheral
system: a diagrammatical one, in terms of link diagrams, seen as welded
diagrams up to self-virtualization, and a topological one, in terms of ribbon
solid tori in 4--space up to ribbon link-homotopy.
