A Phobia or Fear of hats or headgear, perhaps what is under them, or of wearing them.
The opposite of this would be the love or fetish of hats kapelophile which likely stems from the
French words chapeau, chapeaux, chapeaus, capel or chapel which originate from the Latin word cappellum "
hood,
hat", the diminutive of cappa "hooded cloak", which is also where
cap pulls it's original meaning from.
Thus, you could extrapolate kapelophobia or chapeauphobia, chapeauphile could just as well be valid words.. if you are creative like that, and they sound a more appropriate than... cock-laphobia
We seem to be lacking good long fancy words for
hat, which is a real shame. *tips
hat*