Are you certain it's glued directly to the screen? On our Jazz it is removed by lifting a clip and the mirror slips off leaving the spigot behind. The Civic re view mirror on some models is the same. I have a rear view mirror for the passenger and that is a Honda rear view mirror mounted on a 'spigot' which is glued to the screen.
Have a good look, you have to be a contortionist as you have to look upwards at the back of the mounting arm. Try a local friendly windscreen place. if you are worried about doing it.