The filter was supposed to be circular, but who knows. Is there any way to tell?
Sure---and it's easy. Look through the filter into a mirror at the filter's mirror image. Flip the filter back and forth so you'd look through it once from the front side and then from the back side. With a linear polarizer, the filter's mirror image as seen through the filter won't change; the filter always will look grey. With a circular polarizer, the filter's mirror image will change from grey to pitch-black when flipping it.
When using a linear polarizer on a camera which needs circular polarizers then over-exposure would be the typical thing to happen. And the Sony A350 does need circular polarizers ... as all DSLR cameras do.
Your test works perfectly, and it did what you said. I even compared it with a Hoya polarizer which does indeed turn black. It is definitely a linear polarizer! What a piece of junk! I have bought cheap polarizers before and they always worked although I suspect that their coatings were not very good. I guess its off to the photo store to buy a name brand polarizer and to spend the money I suppose I should have spent to start with.
Thank you to everybody for your help!
Although I suspected it was somehow the polarizer, it did not occur to me that I had been sold a linear polarizer (mail order from an Amazon associated dealer).