It's not terribly uncommon. My wife also has it. It's also related to the length of my hair, where longer hair is significantly worse. Presumably it's the oil build up as my hair is extremely oily and fine.

edit: I am fully aware that not washing leads to less oil build up over time, but I have tried and doctors have tried and that boat has sailed.

I'd imagine a hair wash in the sink also does the job when in a hurry then? Do dry shampoos work too, or does it need a wet wash?

Yes, washing the hair in any capacity works. Unsure about dry shampoo.

I see, that's interesting. Initially sounded like a hair weight issue, so by wet washing it you remove enough sebum to not drag on your scalp too hard. But I'm not sure it adds enough weight for it to do that.

I wonder that over time your scalp produces enough of some chemical to give you a headache, and so you need to rinse it away before that happens? Especially since you mention that any wet wash helps, so the chemicals in the shampoo aren't actually relevant for it. Interesting issue though, don't envy you.

Could be muscle tension or something helped by the warm water