Social media has become a common arena for both far-right and Islamic extremist groups to stoke division through the spreading of propaganda and hate speech. This online hate is suggested to drive extremism online, and in some cases lead to offline hate crimes and violence. Whether this online radicalisation happens in isolation within a group, or whether there is an interdependent relationship of mutual radicalisation, is unclear. A possible process by which mutual radicalisation could occur would be if social media incite users to commit offline violence, and if this offline violence in return triggers online reactions from both the target and perpetrator groups. This however has not been tested. This study addresses these questions by investigating the nature of the online-offline relationship of extremist hate. We combine data from the social media platform Gab, variations in Internet search trends, and offline hate crimes in three countries, and test for temporal relationships between opposing extremist groups. Our findings show that online hate from far-right groups both precedes offline violence from these same groups, and spikes following offline violence from opposing Islamic extremist groups. Additionally, far-right Islamophobic violence offline is also followed by increased online interest in Islamic extremist topics. Together, these findings show that the Internet, and specifically hate speech, plays a potential key role in a cyclical process that increases mutual radicalisation.