Interesting topic! I don't normally have this problem, but you made me realize what happened to me last night. I was riding, and we got a sudden and very intense thunderstorm. I'm in New York City, so I don't think it's necessary to take shelter during these. I kept riding, and one of my eyes started stinging badly. I thought it was a particle. I couldn't get it out with blinking and tearing, so I reluctantly wiped my eye, and that relieved me of the pain. Now I know it was sweat.
I generally wear a cycling cap under my helmet. @
chas58, yes it can make me hotter, but my sweat makes my helmet disgusting, so it's like underwear.
I agree that synthetic is best for summer. The new material which I've heard called "jersey material" is a kind of polyester that feels like nylon. I might try that liner that @mynewnchrome cited. My synthetic summer hat is OK, but maybe I could do better. I have some t-shirts made of polyester, and I swear, wearing them makes me cooler than wearing nothing. It's as if they're made of an air conditioner.
The other ideas offered here are very interesting, too.