Dysfunctions in online social networks (e.g., echo chambers or filter bubbles) are studied by characterizing the opinion of users, for example, as Democrat-or Republican-leaning, or in continuous scales ranging from most liberal to most conservative. Recent studies have stressed the need for studying these phenomena in complex social networks in additional dimensions of social cleavage, including anti-elite polarization and attitudes towards changing cultural issues. The study of social networks in high-dimensional opinion spaces remains challenging in settings such as that of the US, both because of the dominance of a principal liberal-conservative cleavage, and because two-party political systems structure preferences of users and the tools to measure them. This article builds on embedding of social graphs in multi-dimensional ideological spaces and NLP methods to identify additional cleavage dimensions linked to cultural, policy, social, and ideological groups and preferences. Using Twitter social graph data I infer the political stance of nearly 2 million users connected to the political debate in the US for several issue dimensions of public debate. The proposed method shows that it is possible to identify several dimensions structuring social graphs, non-aligned to liberal-conservative divides and related to new emergent social cleavages. These results also shed a new light on ideological scaling methods gaining attention in many disciplines, allowing to identify and test the nature of spatial dimensions mined on social graphs.