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We prove that Szczarba's twisting cochain is comultiplicative. In particular,the induced map from the cobar construction of the chains on a 1-reducedsimplicial set X to the chains on the Kan loop group of X is aquasi-isomorphism of dg bialgebras. We also show that Szczarba's twistedshuffle map is a dgc map connecting a twisted Cartesian product with theassociated twisted tensor product. This gives a natural dgc model for fibrebundles. We apply our results to finite covering spaces and to the Serrespectral sequence.