I had a tiny heureka moment and I thought I'd share.
Te and Fi system and Fe and Ti system are different because...
in Te-Fi, there's a bunch of cold facts which are associated with emotional links: "but that's better and that's bad, I don't like that one". The facts themselves have no emotion. How one fact associates with another one, does have an emotion. Like a bunch of boxes with different characteristing and it needs Fi to compare them according to "one box of prettier than the other box and the third box looks so tiny compared to others."
in Ti-Fe*, there's a bunch of colorful emotional evaluations with very clear and logical associations. "This wonderful box is the smallest one. The nasty and unreliable box is very well studied compared to others". The facts are just evaluations, but their relations are clear. Basically, have you noticed how Ti types have an opinion about everything, even though ENTps have many opposing opinions at the same time and INTjs rarely say what their opinion is.
These are completely different systems which are very difficult to integrate! so when Te and Fe types are talking, they end up arguing, because one talks about the pure evaluation of the facts and the other one talk about how much he likes it really. And Ti and Fi types are talking about different things, as one tries to explain the logical interactions of the facts and the other one is trying to say which facts interact in the nicest way. Te and Ti clash because there ends up being just cold facts with cold clear interactions. Maybe some of the facts should be treated as less valid or less reliable compared to other facts. And Fe and Fi clash because then there would be just evaluations of facts and the evaluations of their relations. That would result with a totally subjective system where nothing can be trusted.
so here. That's some badly explained socionics theory that relates clearly to the definitions of all the four judging functions.
EDIT: It was supposed to be Ti-Fe (not Ti-Te).