Twitter app disappears in iTunes

When I download the official Twitter app to iTunes (10.1.1 on OS X), it appears to download correctly, and I can see the file ~/Music/iTunes/Mobile Applications/Twitter 3.2.1.ipa. However, it doesn't appear in my iTunes library, and doesn't get synced to the iPhone.

So I tried installing the app on the iPhone. This worked fine, until I synced the iPhone to iTunes, whereupon the Twitter App duly disappeared from the phone.

I'm very confused, and the iTunes support people were completely unhelpful, other than to provide the link to this forum and to suggest going to my local Apple store - since I have a Mac Pro, this isn't very practical.

Does anyone have any ideas? Help much appreciated...