Two classes of disagreement
I learned about the concept of two classes of disagreement from an anecdote in Michael Hiltzik's book Dealers of Lightning on the history of Xerox PARC.
He describes strong tensions between Robert Metcalfe and Chuck Thacker, who are both pioneers of the Ethernet and personal computing. Hiltzik quotes Bob Taylor, the manager of PARCs computer science lab, who introduces what he calls a "Class One disagreement" as the root cause.
According to Taylor, two people have a Class One disagreement when they disagree and neither can explain to the other's satisfaction the other's point of view. When each person can explain to the other's satisfaction the other's point of view, it's a Class Two disagreement. The latter enables people to work together, even if they disagree, while the former is destructive.
With that framework in mind, when you "agree to disagree", it would be wise to ensure that it is on a Class Two disagreement.