The "eagle's head always facing forward" makes sense to me - and is what would explain it pointing to the laurel branch on one and the arrows on the other.
Just like an American flag patch or sticker should always have the starfield pointing forward "into the wind" (just like on a flagpole in the wind). It's always bugged me to see a sticker on the right side of a car with the stars on the left side, downwind towards the back of the car. Why they don't make/sell them both ways - but then again most people don't know or don't care.
I've never been in the military, nor claim to be any kind of historian, but it all sounds logical to me.
Edit:
Here's an image I found while searching about flag etiquette: