This study advances our understanding of the heterogeneity of Higher Education Institutions (HEIs) in Europe by leveraging on an extended version of the European Tertiary Education Register (ETER, www.eter-project.com) enriched with research output data within the RISIS2 project (risis2.eu). By using advanced statistical techniques, we are able to classify more than 2,000 HEIs 28 European countries in six classes, that differ in terms of two main dimensions, i.e. the orientation towards education vs. research and the subject specialization towards natural sciences vs. human and social sciences. Further, we show how the classification can be mobilized for comparative analysis of the structure of national higher education systems. On this basis, we distinguish between three types of national systems, based on the role of universities in education and on the extent of specialization in technological research and education. We further demonstrate different patterns of blurring between the universities and the college sector, including a model where colleges develop a distinct specialization towards technical education and applied research. The classification has relevant scholarly, policy and managerial implications. It allows investigating heterogeneity in higher education beyond national specificities and might contribute to a better understanding by policymakers and higher education managers of the diversity of the classes of HEIs found in the system. The latter will be relevant in order to design tailored policies to specific classes of HEIs, but also for HEIs themselves to develop strategies that fit their own characteristics and positioning within the system.