The international trade network (ITN) has received renewed multidisciplinary interest due to recent advances in network theory. However, it is still unclear whether a network approach conveys additional, nontrivial information with respect to traditional international-economics analyses that describe world trade only in terms of local (rst-order) properties. In this and in a companion paper, we employ a recently-proposed randomization method to assess in detail the role that local properties have in shaping higher-order patterns of the ITN in all its possible representations (binary/ weighted, directed/undirected, aggregated/disaggregated) and across several years. Here we show that, remarkably, all the properties of all binary projections of the network can be completely traced back to the degree sequence, which is therefore maximally informative. Our results imply that explaining the observed degree sequence of the ITN, which has not received particular attention in economic theory, should instead become one the main focuses of models of trade.