We provide a systematic decomposition of world trade using harmonized bilateral flows at the most available detail (some 5,000 product categories), into three trade types: inter-industry, intra-industry in horizontally and in vertically differentiated products. The analysis is diachronic and considers country pairs such as France-Germany, United States-China, Malaysia-Singapore, or India-Nigeria. We show that the increase in IIT at the world level is due to two-way trade of vertically differentiated products. We find France and Germany having the highest share of IIT in their bilateral trade among all country pairs in the world. In value terms, the most important bilateral IIT is between the United States and Canada. Recently, specialization according to the classical theories of international trade (inter-industry trade), has recovered, due to the increasing participation of emerging economies in world trade.