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Like the Coxeter graph becoming reattached into the Klein graph in [3], the Levi graphs of the \(9_3\) and \(10_3\) self-dual configurations, known as the Pappus and Desargues (\(k\)-transitive) graphs \(\mathcal{P}\) and \(\mathcal{D}\) (where \(k = 3\)), also admit reattachments of the distance-(\(k – 1\)) graphs of half of their oriented shortest cycles via orientation assignments on their common (\(k – 1\))-arcs, concurrent for \(\mathcal{P}\) and opposite for \(\mathcal{D}\), now into 2 disjoint copies of their corresponding Menger graphs. Here, \(\mathcal{P}\) is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while \(\mathcal{D}\) is one of 7 CDT graphs with the opposite-reattachment behavior, including the Coxeter graph. Thus, \(\mathcal{P}\) and \(\mathcal{D}\) confront each other in these respects, obtained via \(\mathcal{C}\)-ultrahomogeneous graph techniques \cite{4,5} that allow us to characterize the obtained reattachment Menger graphs in the same terms.