To compare with and without polarizer you .need to have the correct exposure in both images. That way the only thing that changes is the effect of the polarizer.
Right. So here are two pictures with full-auto exposure.
With the polarizer rotated to brighten the sky:
With polarizer rotated to darken:
I took those pics a while back so I don’t exactly remember if I just rotated the polarizer away or removed it, but you can clearly see those two pictures are very different. You can also see how the polarizer is not uniform: the left side of the sky, is darker than the right side. I find that effect a little annoying, but it’s something that can be useful at times as well…
I think it’s due to fact that sun is on the left. Birds can use light polarisation to locate sun in cloudy weather.
Most spectacular effects are when using filter with sun at 90°. Getting above grading from left to right is due to wide angel view of the lens so if we assume that the left side is at " 90° " to the sun, the right side is 90° - “angel view of the lens”. With normal and tele lens the difference is less visible. These two bellow aren’t taken at same conditions but shows difference with/without polarizing filter. Effect is amazing, starting from sky, clouds,and pines (reflecting wax on needles) and mostly on water. Photo made with 20mm m43 -> 40mm FF- > circa 50° angel view. So if is the sun is 90° to my right, left and right border of photos are only 65°. Photos are SOC, GX7 20mm/1.7