Step 1: Making the Half-Shape
To make the first half, we will make a half-square one inch on the sides. Do this in a 2-D Sketch as shown. The diagonal will be used as the axis of rotation.
Step 2: Making the Solid of Revolution
Next let's make a solid of revolution from the half square, as shown, by using the Revolve 3D command. Revolve the triangle 180 degrees with the hypotenuse as the axis. You can think of this solid as two half-cones mated at the base.
Step 3: Mate the Two Halves
Make the other half by copying the solid you just Revolved. Note that both sides have a square base. If you mated the square base with the apexes matching, you would get two cones (a "bicone") joined at their base. But here's the twist: to make a sphericon, rotate one of the halves 90 degrees with respect to the other, so the apexes met the base edges. Then mate the two halves by Constraining two of the square edges to be coincident.
Step 4: Printing and Polishing
This turned out to be an easy print on the Objet stereolithography printers at Pier 9. I used the Vero clear resin, and polished it. I started with the polishing instructions from Robb and Noah, but since I am a little lazy, I skipped a lot of the grit and the steps. I started with 320 grit sandpaper, then went directly to a wet sand with 1000 grit. I missed a few spots and had to touch up with the 320 grit again, but NBD. After that I went directly to the buffing wheel. Perhaps not optically perfect but it looks fine to me, and it took perhaps 30 minutes.
Here's another version with smoothed edges in case anybody cares ;)
Downunder35m made it!