You see, both of those popular reasons—the blue skies and the bright reflections—are actually variations on the same answer. Because what polarizers fundamentally do is they eliminate reflections. They eliminate light that has scattered and simply don't let it through the filter and into the lens. So, when a polarizer makes a blue sky bluer, it's really eliminating scattered light from reflections off of moisture in the air and the atmosphere in general. It's cutting through the glare, and revealing the color—just like it does on a reflection on windows or a majestic mountain lake. This is what Polarizers really do, they cut through the glare. And that's why they deliver such great color.
So, once you understand what's really happening with a polarizer, you begin to use the filter whenever you want to ensure you've got rich, accurate colors. It's useful in all sorts of photographic applications, and even—believe it or not—on a cloudy, gray day. True, the sky may not be blue, but everything else in the frame will look richer and more vivid if you use a polarizer filter even on a cloudy day, because it's going to cut through the glare and provide better color.
One of the most common places to see vivid color come through is on trees and foliage. All of those shiny leaves reflect light, which means that while yes, you can see that a tree is green even on a cloudy day, you're really seeing a whole lot of glare and reflections, too. Put on the polarizer, though, and suddenly you eliminate those subtle reflections and the green of the tree leaves shines through—all the way to your camera, which sees reflections and color and anything else you put in front of the camera. So, you've got to be deliberate with what you show it. Polarizers can help you show your camera only the most important colors in a scene.
This same principle holds true on practically every other element besides foliage—like the beautiful tiles on a home's roof, for instance, or the bright color of the brick on the façade. It works on cloudy days and sunny days, it eliminates glare from windows and cars and almost every other surface light bounces off of. If the day is not only gray, but also rainy and drizzly, the effect is even more noticeable. Why? Because all of that moisture coating every visible surface simply creates a lot more opportunity for reflections to rob your scene of rich colors. But with a polarizing filter, those color-blocking reflections all but disappear, and your photos become more vivid and saturated, no matter the subject, and no matter the weather. And, no matter if you didn't even know what a polarizer is really perfect for.