Proverbs 25:21-22 demands that we serve our enemy when he is hungry or thirsty, and in so doing, we heap burning coals on his head. It is a reminder that our actions communicate and accomplish much more than our words alone. We can be great at lip service to an apology and attempts to make things right.
When I serve my enemy, I place them in a position where any further attack on me would be crazy, because what kind of person attacks someone who cares for you?
I can also defuse a situation that is only a situation and allows you to see a person. People deserve respect, and when our frustrations are out of the way, we may actually find someone we like a bit. We often dislike in people the characteristics we dislike most about ourselves.
Reconciliation requires a bold act of humility and is not concerned with payback. It is not concerned with selfish demands. It is not concerned with whether the response will be your view of "equal".
Who are you in an argument with right now? What is some way you could enact a bold act of humility and completely disarm the argument and the person to find reconciliation?