I've finally figured out Fi. It's amazing, really.
It's nothing but Ti in reverse.
To explain:
Ti starts from objective first principles... but subjectively valuates each of those principles. For example, I consider social stability and cohesion to be the basis for any human society, so my logical determinations for social strategies are going to stem from this key conceptualization. Someone who considers voluntary affiliation to be that sort of basis will come to completely different determinations, though these may still logically connect. These determinations will arise from external logical patterns and connections, but they still relate to one another based on a prioritization that stems from theoretically subjective decisions (though Ti users will argue you to death on why their decision is the objectively true one... hint, guys - you're wrong, I'm right, deal with it). We argue these things to death because Fe desperately wants to get others to agree with us.
Fi does the opposite. It starts from the objective conclusion (manifested in several ways, for example, in the relative intelligence of Fi users, and in the "Fi mirroring" many people point out), and then winnows its way to the subjective valuation of each of these. The prioritization hierarchy (aka the tert/inf Te) becomes the endpoint of the Fi judging process, rather than the start of it like Ti. Since we're all human, and go through several of the same experiences, many of these valuations will converge in most xxFPs. However, there's no particular reason they have to, and they often do not.
The thing is, both of these have an emotive and non-affective component to them. The valuation/prioritization is the emotional part. This is demonstrated by the way you genuinely do evoke emotions from a Ti user - attack that person's core logic. "Your conclusions are idiotic" won't do a thing, because we'll simply assume the other person's stupid; however, "your points aren't even worth considering" will get a rise, only because the initial valuation principle will suffer from invalidation.
Notice where I'm getting to?
This is exactly the same sort of reaction Ti users complain of about Fi users when their personal values are offended! Likewise, Fi users do have a non-affective aspect of their judging function - this is how they dauntlessly face all aspects of the world in the first place. Fi seeks these objective conclusions in the first place, without any regard to personal impact. It wants to universalize its valuation/prioritization through experience in much the same way Ti wants to universalize its possible range of conclusion through logical analysis.
They're really two sides of the same coin, one edge being "subjective" and the other "objective".