Hmm... Interesting. In doing more research on the RDA5807 chip sets, it appears that they have to modes. One to emulate the TEA5767 chip (which is the mode for which I've been writing code) and a separate RDA5807 mode, which has more capability. It explains the different I2C addresses (0x10/0x11 for RDA mode, 0x60 for TEA mode).
What this means is that I'm not yet done with the breakout board and I'll probably be spending days, trying to find an English-language data sheet which includes explanations of the various registers. About the only work that's been done with the RDA mode appears to have been in the Arduino community. Will keep y'all posted.
(updated 16 Mar to fix spelling)