To be sure my setup was working, I pulled out the Star Wars 3D movie and loaded it. It came up in 3D. So it's the disc.
A bit later, I put the Avatar disc back in and lo and behold, IT came up in 3D. So there is something that changing discs did to get the system into the right state to play the movie. Very inconvenient, having to mess around for almost 30 minutes before finally getting this to work. Seems random!
I've already blasted an e-mail to Oppo about this.