I have integrity, he's a stubborn ass

Teddy bears have integrity, but so do pit vipers. We like to think it's a good quality, and that we ourselves have it. In that sense it's not a very useful concept, since most of us are fully capable of convincing ourselves of pretty much anything, including imaginary virtues.

Integrity, to me, includes applying the same standards to yourself that you do to others, that is, the opposite of hypocrisy. But that takes insight, and there are plenty of people who fall within the teddy bear/pit viper definition of integrity who don't really have much in the way of insight. That is, you can trust them to be themselves at all times; and to stick with their principles under fire, even if you don't agree with their choices.

It's hard to consider integrity a virtue unless it's driven by some kind of internal governor. So if I were to say unironically who I thought had integrity, it's going to be based on my perception of what constitutes their internal governor (that set of principles which drives them) but also to some extent, what my judgment is of those principles. So I can be wrong on two levels: both in what I think drives this person, and in my judgment of those drivers. Well, I can also be wrong about how they live up to those drivers. If I leave my judgment out of it (as if one really can), I can define integrity a little more loosely. I like to call it the Ordell Robbie theory:

You can't trust Melanie but you can trust Melanie to be Melanie.