I dropped my CPL onto some rocks and it's a bit scratched up. There's no obvious effect in a (very cursory) look at the photos, but are there tests I can do to see if the damage has a visible effect?
The end of the lens is extremely out of focus, it takes a lot of damage to be immediately visible. One test I've heard of is to attach a few strips of post-it paper to the front of the lens - you'll barely be able to see the difference.
The most likely effect will be a slight blurring of the image. Take a couple of pictures of a static scene using a tripod, one with the filter and one without. If you can't see the difference, it's time to stop worrying.
You can test the polarizing effect by placing a non-circular polarizing filter in front, you should be able to place it at an angle that almost eliminates light transmission. However I highly doubt that the scratches will have an effect on it's ability to filter polarized light!
As already stated scratches on the front of a lens typically have a minimal impact on image quality: here's a very good example.
Shoot a photo so that a light source is in the frame to test if a damaged filter (or lens front element) has visible effect. Look for extra flare or streaks shaped like the scratch(es).
I've heard of fixing the flare by painting over the scratch with a marker, but haven't had a chance to verify it. You might want to start out with a whiteboard marker to test.
Also, try using big F number which will make image defects more perceptible.