Cartesian to Polar Coordinate transform for a rectangular seamless texture

It’s funny but I can’t describe the issue in the title sufficiently:
I need to distort the UV map such that when I map that texture onto a sphere, it looks like it wasn’t distorted, i.e., it looks like the original square seamless map I started with prior to distorting it for the sphere.

Would anyone be so kind as to tell me how to effect a polar transform of a square (need not be square) seamless texture onto a sphere, that when on the sphere it has no distortion, similar to how they do the textures on spheres as displayed on texture websites, but with the back half not deleted!

I’ve been told I can effect this procedurally, but I don’t even know how to look up the sub subject in procedural modeling to study the issue, so even that I could benefit from.

Thank you for any direction.

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