Got it!! (I think)

The word "whirl" can be substituted for "birl" - ie, "C'mon stales, give it a whirl". Following the thought further, it's only a very small step to thinking of taking one's turn as taking one's chance (in trying to do something). Ergo, we are now considering whirling and birling in the context of a game of chance. Thus, surpise surprise, the expression may have had its origin in the famous Oz game of 2-up (which, as I'm sure you are all aware of, 2 pennies are spun in the air and the outcomes bet upon - 2 heads, 2 tails or a split). The person that flips the coins, the spinner, is giving the pennies a birl. As the spinner's role can be filled by any of the assembled punters, each could say, "I'll give it a birl" as they stepped into the circle.

Another game of chance contender is the "chocolate wheel" - that firm favourite of pubs, school fetes, returned serviceman's ("RSL") clubs etc. You know, the big wooden disk with nails around its perimeter against which a strip of rubber flaps.

Whaddya reckon? I think the "twirl" definition DID crack it - big thanks.

stales