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