I gurgled:> Yow, there are totient(n)/2 of these for every star polygon(n/d), i.e.

(totient(n)/2)^2 for every n. The nine for n=7: www.tweedledum.com/rwg/sphericons7.PNG . No, for even n there are more, because there are two kinds of mirror symmetry.

Oh, foo, I used a needlessly unsymmetrical sphericon to make the baseball, Back to the drawing board. --rwg