Well, I figured it out.

Technical reference manual, section 12.4.4. first stage configuration.

I was writing 16-bit data in Tx FIFO mode1. However, it is being read from the DMA buffer as a 32-bit word, and the 16 high-order bits are output first .
In other words, if your data is an uint16_t array ...

I swear I'm not going crazy.
Basically, I'm trying to use the I2S to output data in parallel. It's not very well documented, but using TobleMiner's library it's not too hard - or so I thought.
It seems to work just fine, except that my data gets corrupted. I've managed to work it down to a very ...