I'd say that whatever cancelling the Right does is in response to the Left. The Right does some cancelling now, to the limited extent they can since they control so few institutions, but over recent years all the cancelling was coming from the Left. The Right started cancelling to the extent they could in response because the alternative was to sit there and do nothing while the Left kept cancelling.The culture wars waged by the right are pretty evident. The cancelling of anything arbitrarily deemed 'woke' is a glaring example.
Twenty or even ten years ago cancelling was not even remotely a thing. It's all recent and it was all initiated by the Left.