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In your image, there is a wide range of angles of reflected light. A polarizer would help with part, but not a lot of the reflections.
In my experience a polarizing filter will work with reflections off any type of surface: metal, glass, water, tree leaves, etc. The greatest effect is on light reflected off a smooth flat surface (glass or water). The major factor is the angle at which the source light is being reflected from the surface. The light is polarized more at steeper angles, so the filter blocks more of the reflection. A direct reflection, such as flash back from camera flashed directly into a mirror, has no polarized rays, so the filter would have no effect.
The same is true with using the polarizer on clear blue sky. The most polarizing occurs in an arc at a 90° angle to the sun. The effect decreases at you move toward 0 or 180°
Circular polarizers are higher priced and have little advantage over the linear polarizing filters. I have both styles and stack them in pairs for variable neutral density (the top one has to be linear). I decided that the circular style was an unnecessary expense since either type works well alone on my dslrs. I have never seen any adverse affects on auto focus, metering or other issues by using my linear polarizer.
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My 2¢ worth
Joe Filer
Mahomet, IL
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This is correct, but this demonstration could still go a little bit astray if the light that is incident on the mirror is polarized. So no one should be surprised if you notice a small effect.It is nothing to do with flat or not. Painted metal is not really a metallic surface (except some paint contains metal flakes of various sizes to create a pearl or sparkly appearance. But the physics of polarizers is such that they do not affect reflections off metallic surfaces - period. This is true whether flat or not. Linear polarizers and Circular polarizers are the same in this regard. \Your first two pictures are self evident, and have nothing to do with anything I said. The third is meaningless because the angle of the camera and knife are not identical, so whether one is darker or not is more to do with that than the polarizer.
If you don't believe me, look at the reflection in a conventional (silvered glass) mirror. The reflection remains no matter what angle you look at it through a polarizer. There will be secondary reflections off the front glass surface, which will be affected by the filter, but the reflection off the silver surface will not.