Summary: In two recent papers [P. R. Hafner, Electron. J. Comb. 11, No. 1, Research paper R77, 33 p. (2004; Zbl 1060.05073); I. Ilić et al., J. Comb. Math. Comb. Comput. 80, 267–275 (2012; Zbl 1242.68101)] it was shown that the Higman-Sims graph \(\Gamma\) can be decomposed into a disjoint union of five double Petersen graphs. In the second of these papers, it was further shown that all such decompositions fall into a single orbit under the action of sporadic simple group \(HS\), which is of index two in the full automorphism group of \(\Gamma\). In this article we prove that the Hall-Janko graph \(\Theta\) can be decomposed into a disjoint union of double co-Petersen graphs. We find all such decompositions, and prove they fall into a single orbit under the action of the sporadic simple group \(J_2=HJ\). The stabilizer in \(J_2\) of such a decomposition is \(D_5\times A_5\). There are striking similarities between the decompositions of \(\Gamma\) and \(\Theta\) just described. Finally, motivated by these decompositions, we obtain new constructions of the Higman-Sims and Hall-Janko graphs from Petersen and co-Petersen graphs.