How to deform a sphere to a half-sphere I want to animate two spheres colliding that end up as one ying yang symbol.

The perfect solution in my eyes would be a way to reverse engineer it: deform a sphere (fig. A) with modifiers to a half piece of the symbol (fig. B).
How to deform/morph a sphere to a half-sphere smootly and suitable for animation?
I tried at first with the Boolean tool, but this is just way too glitchy. I tried Cloth and Softbody simulation too, but this costs too much time. I got the feeling a range of modifiers such as simple deform and displacement could work perfectly. Any clever ideas?
Update 30 min later: I tried procedural modelling with modifiers to it, but it's clearly not so round as I want it to be. Plus there is a weird glitch. See blend file below.
Here is a try:

I've twisted a sphere, kept an edge loop only:

Then used a Screw modifier to make it rotate from 360° to 180°, then gave it Solidify, Bevel and Subdivision Surface modifier: