Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition . For , let be the 1-design in which αIC for αB and if and only if α is adjacent to at least one vertex of C, where is the neighbourhood of B in the quotient graph of Γ relative to . In a natural way the setwise stabilizer G_B of B in G induces a group of automorphisms of . In this paper, we study those graphs Γ such that the actions of G_B on B and are permutationally equivalent, that is, there exists a bijection such that ρ(α^x)=(ρ(α))^x for αB and xG_B. In this case the vertices of Γ can be labelled naturally by the arcs of . By using this labelling technique we analyse and the bipartite subgraph Γ[B,C] induced by adjacent blocks B,C of , and study the influence of them on the structure of Γ. We prove that the class of such graphs Γ is precisely the class of those graphs obtained from G-symmetric graphs Σ and self-paired G-orbits on 3-arcs of Σ by a construction introduced in a recent paper of Li, Praeger and the author, and that Γ can be reconstructed from via this construction.