Originally Posted by Kish
I mean that he has no redeeming qualities. Which isn't quite the same as having no remotely positive moral traits--although considering how warped his concept of "love" in "I do happen to love you" is, that may also be true. He is not capable of understanding why there is anything wrong with his actions; when confronted with actual morality his reaction is, "Huh, that's weird...okay, kill the gladiators then, since my son unaccountably doesn't want to enjoy seeing them horribly tortured."
I believe the example with the gladiators isn't the best one. From his position of a ruthless general, Tarquin had very limited options at that moment. He couldn't tolerate letting badly staged fight to continue (or, that Elan wasn't taking it well), but what could he do instead of giving order to kill them? Could he just send them back to the prison? That would make no sense, people would think him weak or becoming senile, Elan would think he could be easily swayed and Haley would immediately tried to exploit that. And given those gladiators would hate him anyway, even your average Affably Evil guy in his position would just pull the trigger.
Consider strip 854 as a counterexample. After Malack gives him a piece of his mind, Tarquin does seem to realize he has crossed certain lines and apologises. Unless one dismisses that as a staged performance and faked sincerity, it gives a pretty good clue that Tarquin is perfectly capable of understanding when he does wrong - he just doesn't care, or doesn't want to backpedal, as long as it doesn't affect people he actually holds dear.
In my opinion, Tarquin knows what is wrong with his actions, but since he went on the track of a villain and dictator and so on, he just plays it by the script. He is on that way because he wants to live life as a king and go down like a big damn legend. If somebody makes him realise that there are better ways to be remembered, or that his life could give him so much more if he took a different approach to it etc., he may get swayed. Or may not, but it's certainly worth a shot.
Short story: I don't think the comic shows that Tarquin is irredeemable. So far.