Do people really fold up the brim on their hats. I can’t imagine what purpose this would serve other than a sunburned ear.
Or is this just some sort of tourist identification program? 
You’re thinking of a “slouch” hat I guess. They were the uniform of the Diggers in WWI, so it’s more military than tourist. IIRC they are still worn for parade-type occasions.
I recalled not all that correctly: see http://www.awm.gov.au/encyclopedia/slouch_hat.htm and http://www.awm.gov.au/encyclopedia/slouch_army.htm (the former has a picture)
The brims were turned up so that they would not get in the way of a rifle when carried on the shoulder. I think the proper hats only had one side of the brim turned up, corresponding to the side on which the gun was carried.
All right Rayne Man!
That makes some sence.