I'm sorry, but I have to criticize the title of the post before I begin. By what standards does one measure "normal"? As my therapist has been teaching me lately, there probably really is no such thing as "normal" - between Developmental Disorders, Behavior Disorders, Mood Disorders, Intellectual Disability, and Personality Disorders, humanity really is quite fractured when it comes to "normality". Plus, considering that each individual is unique, normality, in my mind, would suggest that at least two people on Earth are not unique in any way whatsoever - that they're pretty much, for all intents and purposes, exactly the same. We might think there are "Neuro-typicals" who are all "normal", but that's just how they appear to us on the outside, exactly the same - we know nothing about the them on the inside until we get to know them more, and realize that they're not all the same.
I also have a hypothesis (nothing new, really) that normality arises from the way a society is structured culturally, ethnically, and ideologically (e.g., autistic people have a hard time in the U.S., U.K., and especially France because a lot of us struggle with language, and it doesn't help that English and French are a couple of the least phonetic languages in the Western World - I was fortunate enough to be the opposite, as I picked up on language very easily, and have even learned some Spanish and Russian; it's just my idiosyncracies and poor conversational skills that get in the way). Also, consider this - black people have rough times in all-white communities, and homosexuals have rough times in all-heterosexual communities. Muslims have hard times in all-Christian communities, and Socialists have hard times in all-Capitalist communities, and so on and so forth. It's all about homogeneity + deviants/outliers and status quo.
Like you, I have a hard time fitting in with both non-autistics (not normal people, because we've established that there's no such thing) and aspies, in spite of the fact that I usually score in the low-mid 30s out of 50 on most AQ tests. As my therapist puts it "I think you're just f***in' weird! But in a good way!" And he can relate to it too, which is the neat thing about our friendship.
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"Works of art make rules; rules do not make works of art."
-- Claude Debussy