The RNase H structural fold defines a large family of nucleic acid metabolizing enzymes that catalyze phosphoryl transfer reactions using two divalent metal ions in the active site. Almost all of these reactions involve only one strand of the nucleic acid substrates. In contrast, cut-and-paste transposases cleave two DNA strands of opposite polarity, which is usually achieved via an elegant hairpin mechanism. In the mariner transposons, the hairpin intermediate is absent and key aspects of the mechanism by which the transposon ends are cleaved remained unknown. Here, we characterize complexes involved prior to catalysis, which define an asymmetric pathway for transpososome assembly. Using mixtures of wild-type and catalytically inactive transposases, we show that all the catalytic steps of transposition occur within the context of a dimeric transpososome. Crucially, we find that each active site of a transposase dimer is responsible for two hydrolysis and one transesterification reaction at the same transposon end. These results provide the first strong evidence that a DDE/D active site can hydrolyze DNA strands of opposite polarity, a mechanism that has rarely been observed with any type of nuclease.