Trying to understand how podcast syncing works in iOS 6
Same thing for me, this upgrade has brought nothing but headaches.
You can still open Podcasts in your Music App by doing a search for the podcast name.
Deleting the Podcasts app from your phone and killing the Music apps process (or reboot phone) will bring the Podcasts menu back to the Music app as well, hurrah!
I believe Apple's logic is as follows. If you want to sync your podcasts through iTunes, don't install the Podcasts app. Use the Music app instead. If you want to untether your podcasts and manage them independently on your iOS device, use the Podcasts app.