Round 3 dudes and dudettes:
I think the adcom's official position is right on the correlation between invitation date and turn-to-under review date - there is none. I think mine (round 2-er) changed like a week later, and got an invitation on 12/22. If you look at earlycareermba, he/she received one like a week after the submission date - so yes, it is random.
FYI, I received an invitation e-mail a day after the status on my app page changed. The invitation you would receive will say "Johnson Interview Invite". Thus, yes, periodically checking your app page does pay off
Round 3-ers shouldn't worry much at this point. Trust me, when earlycareermba got the invitation, all Round 2-ers freaked out.