Basically, for some reason that I have been completely unable to figure out, e-mails sent via their automated e-mail system, which includes e-mails that are sent so that you can confirm changes in account roles *and* domain transfers, don't reach me. Period.
And yet, e-mails sent via human tech support reps reach me just fine. And by "don't reach me", I mean that my mail server logs show no attempts to deliver, or any connection attempts, or anything.
I called them this afternoon, and got some lady who was entirely unhelpful. She kept jumping to conclusions while I was trying to explain what was going on, without waiting for me to finish (which left her jumping to the wrong conclusions) and then she concluded that I needed the domain auth code (required for .org transfers) sent to me. I tried to explain that I already had that, and that I just needed to approve or authorize the currenty pending transfer request. She decided to not hear me, sent me my auth code (which I already had) again, and then hung up. She was rude and annoying.
So, tonight I called back, and I got a guy who spoke rather broken English, but seemed to be fairly polite. After talking to him for a few minutes (he actually listened to me!), he seemed to understand what was going on. He asked if I had a few minutes, and tried resending the automated e-mail that allows me to confirm the transfer. After a couple of minutes, and it not showing up, he said that there were two ways we could proceed. The way he'd prefer, is to copy and paste the automated e-mail into his own tech support e-mail, and sent that to me. That way it gets sent to my domain's contact e-mail address, maintaining proper security authorizations. The other way involved him talking to his supervisor to manually approve the transfer.
I had no problem with the former, since I'd already established that tech support e-mails got through to me fine. So, he copied, pasted, and sent. A minute or two later, the e-mail arrived, and I managed to confirm the transfer request.
I'm finally free of those annoying and overcharging buggers.