I've heard from a friend that they use GitLab's built-in management system [1]. It might not be worth paying them just for this feature though...
1:https://docs.gitlab.com/ee/operations/incident_management/on...
1:https://docs.gitlab.com/ee/operations/incident_management/on...
It does not seem to do more than something like PagerDuty or OpsGenie, in that there is no integrated way to run specific actions on rotation.
But I did not know that GitLab did that, thanks for the link !