Jeez, I'm an idiot.
Why not just have two squares with each holding one of the poles in its center, and laying one on the bottom of the other (ignoring the obvious deformation)?
So like, the edges of the squares would be the equator, and moving east and west would rotate the tiles on the square...
Though to be perfectly honest, at this point I'm beginning to wonder if it's possible to map this with circles.
Yes, using two squares is a very good way to map the sphere as it removes the coordinates singularity which is present with polar representation. You map each half of the sphere with a Stereographic projection from [-1,1]^2. Then you stitch your two mappings on the boundary of [-1,1]^2 (this is a fairly trivial step).
You can do the same with two circles (with polar mapping), but this way you'll loose the absence of coordinates singularity. So in my opinion, using two circles has no advantages.