My wife and I faced this problem early in our marriage. When we disagreed and got a bit heated about it, her powerful need was for us to talk it through and keep talking until it was all agreed and there was peace. My need, just as powerful, was to get off by myself until I could calm down and avoid saying hurtful things that could never be unsaid.
These are both excellent plans of action — unless you are trying to make them both happen at the same time.
Without endorsing even a speck of the Jungian paraphernalia surrounding the Myers-Briggs Type Indicator, I can tell you that when, at the urging of a good LDS friend, my wife and I read the book "Please Understand Me" by Keirsey and Bates, and took the test at the end of the book, it opened our eyes to each other's needs.
Once we realized that the reason our struggles became so bitter was because we both needed a different way to process the disagreement, we have been able to work out compromises: My wife has learned to wait a while sometimes to settle things, while I calm down; and I have learned that when I can, I must talk things through that I would prefer not to discuss at all.