Rifle bolts can be either side depending on if the firer is left or right handed, in fact I think the SA80 is the first weapon that isnt designed for ampidextrose fittings.
Also during drill rifles are carried over the right shoulder a brim that side would mean soldiers knocking their hats off all the time.
Bill it is possible the picture is around the wrong way or maybe, as with so many army traditions and military dress items, it is possible that the flap is up one side and not the other just because a light horsemen at gallipolli or some such put that side up because the sun was on the other side of their trenches - not that I am saying that happened - just that it is worn in such a way because of something that happened in a battle or similar.
My own regiment wear a black tombstone behind their cap badge to commerate the fact that we wear at the battle of Quebec and were with Wolfe when he fell
The Gloucesters where a cap badge on the front and back of their berets because at waterloo they had to fight back to back against french cavarly because they were so out numbered.
So you see army traditions are necessarily things designed for function.