  In a previous paper we proved a result of the type "invariance under
twisting" for Brzezinski's crossed products. In this paper we prove a converse
of this result, obtaining thus a characterization of what we call equivalent
crossed products. As an application, we characterize cross product bialgebras
(in the sense of Bespalov and Drabant) that are equivalent (in a certain sense)
to a given cross product bialgebra in which one of the factors is a bialgebra
and whose coalgebra structure is a tensor product coalgebra.
