  Eisermann and Lamm introduced a notion of symmetric equivalence among
symmetric union diagrams and studied it using a refined form of the Jones
polynomial. We introduced invariants of symmetric equivalence via refined
versions of topological spin models and provided a partial answer to a question
left open by Eisermann and Lamm. In this paper we adopt a new approach to the
symmetric equivalence problem and give a complete answer to the original
question left open by Eisermann and Lamm.
