The picture is what BagginsBill means. A uv mapped sphere in the middle, viewed from the outside. A clone of the sphere, no changes but size and viewed from the inside. Same identical map used on both with no changes. It is the same sphere viewed from the two different perspectives at the same time.

The east/west directions are reversed.
When you take the picture of the mirror ball, you are the on the outside.
When you make the picture into a environment (viewed from the inside), the directions are flipped.

EDIT - forgot to say I had to rotate the small sphere 180 degrees; else you would be looking at the opposite side also.

Friends don't let friends use booleans.