Humans have a machinery reinforcing their world view. Contradicting observations and facts are usually ignored or interpreted such that it matches their already held opinions.
So for example, it's impossible change the ranking of brands in peoples' minds, in a given category (not absolutely impossible, but extremely hard). But what's possible, is create your own category, and own it.
The book gives the example of 7UP which famously branded itself as the "Uncola". It could do that because it was clear instead of dark; but the point was to create a different category instead of trying to fight Coca-Cola on its own turf.
I think this works in many other fields. People who fight against oppression are good; but if you label them "terrorists" they're bad -- worse than bad: they're not really people any more.
Politics is not a game of facts, it's a game of labels.
[0] Positioning: The Battle for Your Mind (1981)
That being said, having shared experience can help, at least anecdotally, in being able to come to terms with wildly differing views. One example I have is with my brother. With the same event we had come away with a drastically different understanding. The facts between us all checked out, and we confirmed it many times. Yet the shared vocabulary gave us a way of acknowledging the others disagreement in the shared experience.