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Yeah, the only difference in connotation I get from "headwear" and "headgear" is that the former seems more casual and everyday, while the latter seems more formal, or even relating more to a sports...

In that situation, I think using the word "headwear" would be perfectly fine. Since things you mentioned like turbans wouldn't really fit well under a "hat" category, I think headwear is fine, as it...