I solved my own problem! I modified the EmulationStation startup script to toggle back and forth between horizontal and vertical mode on a restart. So, now I can rotate my monitor, restart EmulationStation, and my menus match the orientation. Here is the updated file:
/opt/retropie/supplementary/emulationstation/emulationstation.sh
#!/bin/sh
esdir="$(dirname $0)"
while true; do
rm -f /tmp/es-restart /tmp/es-sysrestart /tmp/es-shutdown
"$esdir/emulationstation" "$@" $rotate
ret=$?
if [ -f /tmp/es-restart ]; then
if [ -z "$rotate" ]
then
rotate=”--screenrotate 1”
else
rotate=””
fi
continue
fi
if [ -f /tmp/es-sysrestart ]; then
rm -f /tmp/es-sysrestart
sudo reboot
break
fi
if [ -f /tmp/es-shutdown ]; then
rm -f /tmp/es-shutdown
sudo poweroff
break
fi
break
done
exit $ret