I was going to say 'On his head.' but I guess that won't help. The fact that the 'digger hat' has traitionally had an easy to manipulate hook and eye arrangement suggests that 'Either way.' is the correct answer. Without doing any research but relying on my razor sharp [63 year old] memory, I'm going to suggest that 'turned up' is the preferred modern style, which one sees in photos of parades in particular, but that clearly 'down' was the more practical method in sun and rain and seems commoner the further back one goes in time, especially in 'action photos'.
All obvious, I know, but my point is that the further back one goes, the less rigidly the regs appear to have been enforced in any but formal contexts: photos of the Aussie Light Horse in the Middle east, for example, and of WWII troops in the Pacific, show them worn down, for obvious practical reasons. Photos of individuals in portraits and so on more often seem to show them up - 'for the look of the thing'. That said, one photo, which I will attempt to locate, shows light horse - near Beersheba, I think - with most brims down but one or two up, supporting my thesis that at some point it became a matter of individual preference. As someone who wears an Aussie style hat most summers, I can testify that 'brim down' wears better, as the weight balances out and the hat seems less likely to slip on one's head.
My tuppence worth and then some!
Peter