1. We draw n points, A, B, C, ... Z, on the upper hemisphere of a sphere, and their n antipodal points on the lower hemisphere, a, b, c, ..., z.
2. We draw the n(n-1)/2 great circles connecting each pair of points on the upper hemisphere, noting that every great circle through X also passes through x.
3. We now slice the sphere into two hemispheres by drawing an arbitrary great circle that does not pass through any of our points
4. Each hemisphere must contain exactly one member from each of the sets 
{A, a}, {B, b}, {C, c}, ..., {Z, z}. We identify each hemisphere by the points it contains (something like AbcDEF); distances between the points are irrelevant.

The question is, How many distinct hemispheres are possible for any given initial configuration of n points? The answer appears to be n(n-1)+2, but I can't prove it.