The theory of echo chambers, which suggests that online political discussions take place in conditions of ideological homogeneity, has recently gained popularity as an explanation for patterns of political polarization and radicalization observed in many democratic countries. However, while micro-level experimental work has shown evidence that individuals may gravitate towards information that supports their beliefs, recent macro-level studies have cast doubt on whether this tendency generates echo chambers in practice, instead suggesting that cross-cutting exposures are a common feature of digital life. In this article, we offer an explanation for these diverging results. Building on cognitive dissonance theory, and making use of observational trace data taken from an online white nationalist website, we explore how individuals in an ideological 'echo chamber' engage with opposing viewpoints. We show that this type of exposure, far from being detrimental to radical online discussions, is actually a core feature of such spaces that encourages people to stay engaged. The most common 'echoes' in this echo chamber are in fact the sound of opposing viewpoints being undermined and marginalized. Hence echo chambers exist not only in spite of but thanks to the unifying presence of oppositional viewpoints. We conclude with reflections on policy implications of our study for those seeking to promote a more moderate political internet.