I suspect that they are given a task within the cult, or they offer to complete some task, and if they fail, then they are punched in the left eye.
In the case of the pope, since his master is Lucifer himself and Lucifer has no physical fist, he self-punishes. We know he got his black eye on a certain trip when he "fell" and hit the frame of his popemobile. Anyone know what might be the failed task?