Political polarization on the digital sphere poses a real challenge to many democracies around the world. Although the issue has received some scholarly attention, there is a need to improve the conceptual precision in the increasingly blurry debate. The use of computational communication science approaches allows us to track political conversations in a fine-grained manner within their natural settings – the realm of interactive social media. The present study combines different algorithmic approaches to studying social media data in order to capture both the interactional structure and content of dynamic political talk online. We conducted an analysis of political polarization across social media platforms (analyzing Facebook, Twitter, and WhatsApp) over 16 months, with close to a quarter million online contributions regarding a political controversy in Israel. Our comprehensive measurement of interactive political talk enables us to address three key aspects of political polarization: (1) interactional polarization – homophilic versus heterophilic user interactions; (2) positional polarization – the positions expressed, and (3) affective polarization – the emotions and attitudes expressed. Our findings indicate that political polarization on social media cannot be conceptualized as a unified phenomenon, as there are significant cross-platform differences. While interactions on Twitter largely conform to established expectations (homophilic interaction patterns, aggravating positional polarization, pronounced inter-group hostility), on WhatsApp, de-polarization occurred over time. Surprisingly, Facebook was found to be the least homophilic platform in terms of interactions, positions, and emotions expressed. Our analysis points to key conceptual distinctions and raises important questions about the drivers and dynamics of political polarization online.