Double pushout (algebraic) graph rewriting, which works by first removing the part of the graph to be regarded as garbage, and then gluing in the new part of the graph, is contrasted with term graph rewriting, which works by first gluing in the new part of the graph (the contractum) and performing redirections, and then removing garbage. It is shown that in the algebraic framework these two strategies can be reconciled. This is done by finding a natural analogue of the contractum in the algebraic framework, which requires the reformulation of the customary double pushout construction. The new and old algebraic constructions coexist within a pushout cube. In this, the usual "outward" form of the double pushout appears as the two rear squares, and the alternative "inward" formulation as the two front squares. The two formulations are entirely equivalent in the world of double pushout graph rewriting. © 1995.