I’m a bit confused. You said above that you cannot reproduce the same effect, but your image comments say the conversion to polar and back works as expected.
Near the top of your image you say “Rectangular to Polar works” but looks like the input to that function was not the rectangular coordates you mention (which is actually Equirectangular or LongLat if I understand correctly) but standard UVs which you could think of as a 2D coordinate plane.
Conversion from 2D UV space to Equirectangular / LongLat can be done in UE4 with the material functions “UV to LongLat” and vice versa.
I wouldn’t really recommend using polar coordinates because the resolution distribution will be terrible. But there is a function “Vector to Radial Value” that can return radial coordinates, but I’d be surprised if it did the same thing as photoshop. Even in your example image, you can see that your ‘polar’ coordinates image doesn’t match the one on the left as there are a different number of radial rings. Maybe its limiting to the upper hemisphere? Maybe multiply G by 2 and see if it helps (just for testing as again I don’t recommend you encode that way).
A better option than any of these methods is Octahedron layouts. Much cheaper to convert between the space. There are material functions for that. You could import a regular cubemap and then use a render target pass to convert it to a single 2d octahedron. Then you’d have a texture thats cheaper than a regular cubemap to use because it wouldn’t have to map to 6 separate faces under the hood (and much cheaper than a longlat conversion since there is no trig involved). I am planning to do a detailed tutorial on this method for my blog soon, since octahedra have many uses.
Just for completeness sake, there is a free photoshop filter called Flexify2 that can do all kinds of space conversions for cubemaps.