This is a series of photos showing what a lens polarizer does over one revolution, in 8 shot increments. The difference from one 45 degree increment to another can be pretty dramatic at a few of the increments, so maybe 22.5 degree increments would be something to try in the future. But this should give you a good idea as it stands.
The photographs were taken off my back deck, across the inlet to all the bays to the South [left of frame] of me [seen here is Huntington bay] around 4pm. They were cropped to 35% of frame, cropping out more of the foreground trees in my backyard above, to the left, and on the bottom of the frame. Focus point was the Lighthouse which is about 2 miles away. The houses in the background are another 1/4 mile further.
Used was my K10D with Sigma 300mm f/2.8 EX DG and the rear drop-in Sigma polarizer supplied with the lens when I purchased it. No TC was used. All shots were taken handheld in Av mode at f/6.3, ISO200. Shutter speed was varied by the camera to keep exposure the same, based on how the polarizer reduced the light more or less as I rotated it around 360 degrees.
All shots were taken within about two minutes, with the same lighting conditions. First shot was with the guide/dot on top of the lens. Next shot was after I turned the polarizer's wheel 45 degrees to the left. Then so on and so on until I made one full revolution in 8 shots.
PP of RAWS done in Lightroom. Landscape sharpening with increased detail setting, high contrast plus some add'l but subtle contrast & brightness curve adjustment. MOST IMPORANTLY, THE WB THE CAMERA CHOSE WHEN IN AUTO MODE, WAS SET TO 'DAYLIGHT' FOR ALL. Use of a Polarizer will cause the camera to change the WB selected, and magnify any color cast/alterations the Polarizer creates; most noticalbe if you use the unaltered JPGs out of the camera. When I cropped all the images to the same size, I also straightened and re-positioned them the best I could. ...I'll have to use my tripod next time. Upon export I resized to 1600 pixels on the long end. They are being shared here resized by the Hosting service Fotki to be 800px wide.
Aside from the obvious color cast change use of a polarizer brings, NOTICE THE REFLECTION OFF THE ROOFTOPS OF THE HOUSES TO THE RIGHT OF AND UP FROM CENTER. The latter is perhaps its most important attribute.
Hope this helps understanding what a poalizer can do for you.