A growing number of social media studies in the U.S. rely on the characterization of the opinion of individual users, for example, as Democrat-or Republican-leaning, or in continuous scales ranging from most liberal to most conservative. Recent works have shown, however, that additional opinion dimensions, for instance measuring attitudes towards elites, institutions, or cultural change, are also relevant for understanding socio-informational phenomena on social platforms and in politics in general. The study of social networks in high-dimensional opinion spaces remains challenging in the US, both because of the relative dominance of a principal liberal-conservative dimension in observed phenomena, and because two-party political systems structure both the preferences of users and the tools to measure them. This article leverages graph embedding in multi-dimensional latent opinion spaces and text analysis to propose a method to identify additional opinion dimensions linked to cultural, policy, social, and ideological groups and preferences. Using Twitter social graph data we infer the political stance of nearly 2 million users connected to the political debate in the U.S. for several issue dimensions of public debate. We show that it is possible to identify several new dimensions structuring social graphs, non-aligned with the classic liberal-conservative dimension. We also show how the social graph is polarized to different degrees along these newfound dimensions, leveraging multi-modality measures in opinion space. These results shed a new light on ideal point estimation methods gaining attention in social media studies, showing that they cannot always assume to capture liberal-conservative divides in single-dimensional models.