I was trying to picture in my head a contrary hatband, and just couldn't do it.
From the economist.com
Wary of losing the silver-trimmed hatband, he slipped it around his neck.
From the en.wikipedia.org
At first they would stick the feather in the adjusting hatband.
From the en.wikipedia.org
The wind picked up, and to keep his silver hatband safe, Cedarstaff looped it around his neck.
From the npr.org
So he gathered things up and put the hatband over his head.
From the sacbee.com
Although a card with Wetherington's name and phone number were in the hatband, the driver had not called to say he had it.
From the newsobserver.com
The hatband separated the crown from the brim.
From the en.wikipedia.org
Another well-known Western accessory, the bolo tie, was a pioneer invention reputedly made from an expensive hatband.
From the en.wikipedia.org
Another sibling, Alexandra, took the hatband business and turned it into RibbonTrim, which imports and distributes fine ribbons, braids and trims.
From the theaustralian.com.au
More examples
A band around the crown of a hat just above the brim
A hat is a head covering. It may be worn for protection against the elements, for ceremonial or religious reasons, for safety, or as a fashion accessory. In the past, hats were an indicator of social status. In the military, they may denote rank and regiment.
The ribbon at the base of the crown. This band normally has a finished bow on the left side of the crown