Matt does make an excellent point. Beta Theta Pi is one fraternity that originally got it wrong when they had a crescent on their badge.
"In 1841 Alexander Paddock (Miami, 1841) improved the badge by making its surface of black eneamel and replacing the crescent by the wreath and diamond. As to the crescent it might be recorded that one reason for the change was that, as borne on the Beta emblem, its horns were turned to the right as one faces the badge. Some Miami student, knowing more heraldry than his mates, or being a closer observer of the devices in the almanac, pointed out that the Beta badge bore the waning crescent, and ridicule hastened the change." ---The Story of Beta Theta Pi, Francis W. Shepardson (1927)
Ours looks just like the one I can see out my window.
ZAX,
Jono