a) protect me from the rain - if raining
b) act as a sunshade - if sunny
I bought a Peter Storm rainhat, but quickly discovered that it would flap up if I was going quickly downhill, or if it was very windy. I decided to modify it. To do that I got an old sunvisor of mine and covered it in a similar coloured material then sewed it on to the underside of the rainhat (see pics). This stiffened it up nicely and it lasted the whole journey and worked very well - so less of the mockery please!
