Absolute value being an odd function

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Correct me if I am wrong, but I learned that for a function to be symmetrical to the origin, it can be rotated 180 degrees and still appear the same. How is ${x^2 - y^2 = 0}$ an odd function if when it is roatated 180 degrees it appears upside-down?

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This is not a function because there are two outputs for one input, so since this is not a function it doesn't obey these rules. Trust the King.