International tax competition is generally framed as states competing for foreign direct invest- ment (FDI), and analyses of the phenomenon draw heavily on FDI statistics. In and of themselves, however, FDI statistics are merely a quantification of the value of investment projects and tell us little about the heterogeneity of these projects and the distinct patterns of competitive dynamics between countries they generate. In this paper, we create a more sophisticated understanding of international tax competition by pointing out its variegated nature. To do so, we introduce the notion of the "great fragmentation of the firm" to distinguish between five categories of FDI: manufacturing affiliates, shared service centers, research and development facilities, intermediate holding companies, and top holding companies. Using a novel combination of firm-level and country-level data, we identify for each category of FDI which European Union member states are most successful in attracting it, what macro-institutional and tax arrangements they rely on for doing so, and what benefits they receive from it in terms of tax revenues and employment creation. In this way we were able to identify five distinct FDI attraction profiles and show that, rather than being a game of all against all, tax competition in the European Union increasingly takes place amongst subsets of countries that compete for similar categories of FDI.