It turns out my root filesystem is on a bcache-backed device, so I need to use
a 
3.10 or newer kernel - AFAICS the latest stable gentoo-sources kernel is
3.8.13.

FWIW my VIDEO_CARDS did include radeon, but not radeonsi. I interpret this bit
of
  /usr/portage/profiles/desc/video_cards.desc
to mean that including "radeonsi" restricts the build to only support SI, not
add support for it:

radeon - VIDEO_CARDS setting to build driver for ATI radeon video cards
radeonsi - VIDEO_CARDS setting to build only Southern Islands based chips code
for radeon

Am I reading that wrong?

The Xorg log I posted shows the radeon driver is being used, and probed my card
as 
having a VERDE chipset.

I didn't file a Gentoo bug because I thought that this part of what I reported
in my dmesg:

[   11.109017] [drm:r600_uvd_init] *ERROR* UVD not responding, giving up!!!
[   11.109588] [drm:si_startup] *ERROR* radeon: failed initializing UVD (-1).

could have tickled a bug that prevents my card from reporting it is
direct-rendering
capable. Isn't this the place to report possible bugs in the DRI code in the
upstream
kernel?

So, please don't resolve this as fixed just yet.