Bak, I didn't use the term edge grain in my response, just stiffest orientation. The parallel grain of the arrow positioned perpendicular to the shelf is not always the stiffest, but it usually is. Regardless, flipping the arrow 180 degrees, which is what happens when cock feather out is nocked cock feather in, would not change the spine whatever it is.
If the arrow was made without ensuring that the stiffest orientation was toward the bow/sideplate, then rotating it some, say 90 degrees or so, could increase the spine as Graps suggests. And, since a lot of folks build arrows without the benefit of a spine tester, and they don't roll the shaft with one end held up, the other on a flat surface and pressure on the center, which is a way to determine the stiffest orientation, there are probably a lot of arrows like that currently in service.
Not sure I understand the reference to self bows. There the edge grain is typically on the side of the limbs. The flat grain forms the back and belly. In effect, the bow bends opposite of the way an arrow flexes at release, though an arrow does flex in multiple directions as it leaves the bow.