The headover is the modern equivalent to the venerable cap comforter used during both World Wars and for decades afterwards. This hat consists of a green fabric tube, open at both ends:
A label is sewn to one end:
The head is passed through the tube which can then be worn around the neck as a scarf:
Or pulled up to make a balaclava:
In design, the headover is probably closer to the 1980s civilian snood than it is to the old cap comforter and it is not possible to easily wear the headover as a simple cap as the old cap comforter could be worn. Despite that, the headover is a small and lightweight item of clothing that can easily be tucked in a pocket and pulled on when required to help protect the wearer from the cold on exercise. It is not universally liked due to its lack of sartorial elegance, however seems to be a simple and practical item.
Tales from the Supply Depot 
” it is not possible to easily wear the headover as a simple cap as the old cap comforter could be worn”
Really?
We used to do it all the time – Fold partway one side, then fold half that part the other side, before folding the lower edge up. It takes a bit of trial and error, but so long as your first two folds result in equal material either side, you end up with something that looks like a chip-bag hat.
Tuck the points in to avoid comments about Thunderbirds!