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Stuart_Armstrong comments on No independence of irrelevant alternatives (picture proof) - Less Wrong
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The axis of symmetry is a property of the figure (in this case, the set of points), not of the axis. In fact, ignore the axis: they don't exist, only their directions have mathematical meaning (neither their scale nor their points of origin mean anything, because the affine transformations of the utility functions will shift those).
Right- let's just call the points A, B, C, -B, and -A. C is going to be the middle of that set, even if you remove (B and -B) and/or (A and -A). When you remove (-B and -A), you've broken the symmetry of the set, even though the set has new symmetry by virtue of there being a new middle point.
If you're postulating a (much much) weaker version of IIA saying something like "if you remove symmetric irrelevant points from a symmetric set, the outcome doesn't change" then you'd be right. But IIA does not require that symmetry be preserved.
Yep, I agree that strong IIA of "if x is chosen from T, and S is a subset of T, then x is chosen from S" doesn't apply if preferences are based on the relative merits of x rather than the individual merits of x. That statement seems obviously true on its own, and so I think the picture proof of this particular example detracts more than it adds, because there is a natural weak IIA here.