The prevalent perspective in quantitative research on opinion dynamics flattens the landscape of the online political discourse into a traditional left--right dichotomy. While this approach helps simplify the analysis and modeling effort, it also neglects the intrinsic multidimensional richness of ideologies. In this study, we analyze social interactions on Reddit, under the lens of a multi-dimensional ideological framework: the political compass. We examine over 8 million comments posted on the subreddits /r/PoliticalCompass and /r/PoliticalCompassMemes during 2020--2022. By leveraging their self-declarations, we disentangle the ideological dimensions of users into economic (left--right) and social (libertarian--authoritarian) axes. In addition, we characterize users by their demographic attributes (age, gender, and affluence). We find significant homophily for interactions along the social axis of the political compass and demographic attributes. Compared to a null model, interactions among individuals of similar ideology surpass expectations by 6%. In contrast, we uncover a significant heterophily along the economic axis: left/right interactions exceed expectations by 10%. Furthermore, heterophilic interactions are characterized by a higher language toxicity than homophilic interactions, which hints at a conflictual discourse between every opposite ideology. Our results help reconcile apparent contradictions in recent literature, which found a superposition of homophilic and heterophilic interactions in online political discussions. By disentangling such interactions into the economic and social axes we pave the way for a deeper understanding of opinion dynamics on social media.