Sick again, so probably just this one post today. It is intended only as a helpful clarification. Your relic find isn't a GAR "medal" (something awarded to a soldier by the Military or Government or organization, for bravery in combat, or honorable service, etc). It is a GAR Annual Reunion "badge," sold at the reunions as a souvenir.
For anybody here who doesn't already know... in this case, the initials GAR stand for Grand Army of the Republic, the yankee equivalent of the Confederate Veterans organization.