Dear Yuriy, if there is no natural mapping, you can always restrict some constraints and generalise the concept. Here is an article by a friend of mine that you may want to go through, to get some insight on spheres and projections. :))

http://www.cmi.ac.in/~anirbit/unispherecopy.pdf

Dear Sonam, thanks for reference. I am pleased to read it. I wonder existence of a continuous mapping or involution.