Sure he can work both ways but only if the writers involved in a "more Punisher type" of character actually had some firm direction they wanted to take Jason in besides being some kind of one note villain for Batman and if they wanted to make him a 3-dimensional character while doing so instead of the 1-dimensional character we ended up with pre-reboot. That's where they went wrong with that sort of angle before the reboot and why the character began to stagnate. All they ever did was tell the same story with different players each time because they hadn't established any sort of clear direction for him beyond the"hey, lets turn him into a villain with daddy issues" angle they were running with back then.