Aha. In trying to use --reinstall, I ended up hitting another bug, #859188. On that bug, commenter Sal Mazzola (@salmaz) suggests using this alternative method to work around it:
apt-get download libgl1-mesa-glx libgl1-mesa-dri dpkg --install libgl1-mesa-*.deb
I now have working GLX again!
Aha. In trying to use --reinstall, I ended up hitting another bug, #859188. On that bug, commenter Sal Mazzola (@salmaz) suggests using this alternative method to work around it:
apt-get download libgl1-mesa-glx libgl1-mesa-dri
dpkg --install libgl1-mesa-*.deb
I now have working GLX again!