I'm just guessing, but if you do the technique classically, without forcing uke to move around you, it seems you generally turn 270 degrees. If you started facing north, you'd end up facing all four directions before you are finished.
As an aside, Peter Ralston's art, Cheng Hsin, has many Aikido techniques in it, yet he calls them made up metaphorical or practical names. They are kind of interesting to hear after being used to Japanese names with little intuitive meaning to me. Shihonage is called "Spinning Brush Stroke", taking into account the spin and the throwing motion.