I'm going to have to disagree with you. Flags are meant to be attached at two points to a flag pole or staff.

A diamond shape flag would not be easy to fly as such. How would you attach it so that it flies as a diamond?

In this case, it appears that the flag is attached to it's staff by the lower staff edge fitting, and unattached to the upper fitting, the staff being
attached to the wall at an angle, about 45°. Thus, it is appears to be a diamond in this case. Rotate the flag 45° counter-clock wise, and you'll
see the inner shape is that of a diamond. At least, that is what it appears to me.

Perhaps it was unattached at the top so that it would appear in a flat configuration, and not in sometime of a heaped shape.

See: http://en.wikipedia.org/wiki/American_Civil_War_Corps_Badges#XIV_Corps.2C_Army_of_the_Cumberland