It works even if the vertices edges and faces aren't straight so what you just described is also equivalent to two hemispheres. Now maybe you've started wonder

It works even if the vertices edges and faces aren't straight so what you just described is also equivalent to two hemispheres. Now maybe you've started wondering where the catch is, since it clearly doesn't work if you just use 1 face for everything. The rule is that the pieces need to be 'simple' which basically means that they're like a disc or simplex, they're allowed to be deformed but you can't cut holes or do anything that changes the topology.