Never Apologize, Never Explain is a tactic adopted by many, indeed it is one of the 48 Laws of Power. The logic goes that by apologizing and explaining your actions your power quotient is inevitably weakened. It is better to let the healing nature of time elapse to wash away the hurt. Change your behaviour if you believe you were wrong and your actions required some justification.
If the game you want to play is power, this might be good advice. If you believe that cooperation is more fruitful than power games, apologizing can be a powerful tool.
Thus playing the PowerGame means
I'm sorry for apologizing.
I'm sorry you feel that way, but I'm not sorry for apologizing.
An interesting perspective to mull: NonviolentCommunication suggests that apologies inherently contribute to a power differential; however, you can instead note what you did, tell how you feel and why, and work out strategies for the future. If done sincerely, this will usually work well in the place of an apology. e.g. "When I now recall how I interrupted you, I'm feeling a bit regretful because I value keeping our dialog open. Can we ... ?" This is without the suggestion that someone was wrong, bad, or unworthy. (Although it may be heard as an apology. That's another story...) --JasonFelice