Bartok And Symmetry Question

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I think this might become clear if you ask yourself a related question: What's the point of using (unusual and complex) timbre combinations when the ear can't distinguish the instruments which make them up?

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 Now, I don't know about anybody else, but I find that there is absolutely no way to tell, aurally, whether such-and-such a note collection is organized around this axis and such-and-such is not.  I was wondering what the point is of presenting note collections that are related through an axis of symmetry when the ear cannot distinguish relationships between them like it can with, say, closely related keys in the tonal system?

The answer is betrayed by your opening statement that you don't know about anybody else. It's as if by the time you ask the question you've forgotten that you said it.

Bartok often uses symmetry to create a tonal structure with a function analogous to the diatonic tertian structure of the common practice. The fact that your ears can't ascertain coherence from this structure does not negate it, Bartok could discern coherence from these structures.

Personally, symmetrical structures are easily recognizable and striking, and to my ears can tonicize a note particularly well with a quality unachievable with diatonicism.