I don't think they were made to go over a saddle because the lettering and decoration is right where the stirrup leathers would hang. It would cause premature wear to the stirrup leather, possibly knock off the brass, and the rider's leg would cover it anyway. In the mid-19th century a lot of Army medical stuff was marked "MD" for Medical Department so that's a possibility. If they were made to hang over hames, they may have been meant to identify an ambulance. In the Civil War period there was no standardized way of marking ambulances and each unit pretty much made it up as they went along. This is extremely speculative, of course.