You’re exactly right, the geometry was not perfectly vertical. I now see that the tool has been designed so that with angled measurements the display will adjust so that it reads “upright”, and that means the display will flip as the angle of the measurement crosses vertical. The problem is there is no tolerance that I can see, or at least it is well below the file tolerance settings as you can see here.
I guess what I really want is a way to specify which way the dimension will display (in the Properties Overrides, under Text Alignment), partly to quickly fix it when there are minor geometry errors, and partly so I can have all the dimensions match when the geometry is intentionally varied but close to vertical. The first case can be solved by switching to the Linear Dimension tool, but not the second without loss of accuracy.
Mostly I suppose I just like using one tool for all my dimensions, and if not for this issue I probably could.