Why is one limited to one only? Assuming that the magic camera fairly will only give me one, than it would depend on the time of day and position of the sun relative to my shooting position. It's noon and the sun's directly overhead, the CPOL isn't going to do diddly-squat; if the sun is lower in the sky and ~90ish degrees to the lens axis, then I'll go CPOL, since I can always hold my hand over the lens as a temporary shade.
well I guess you could use your hand to shield it from direct sunlight like a lens hood would.
So I would use the CPL and my hand as a lens hood.
Just in case your lens hood wouldn't fit if you put on a CPL ?
weird *only* one if question as I would use BOTH in that situation dependent upon location/direction of the sun to the camera.
On sunny days around water I use a cpl to avoid reflections and darken the sky. You can put the hood on backwards, adjust your cpl, then turn the hood back around. OK, if you are taking action shots and don't have time to play with the hood. Leave it off. On most of my lenses I can get my hand inside the hood and spin the filter with a finger nail as my Hoya cir-pl have a serrated type edge.
As you've no-doubt picked up on based on the replies... using one doesn't preclude using the other. Also... they do completely different things.
The polarizer kills reflections on your subject.
The hood protects the lens from direct sunlight which can be useful for reducing flare.
Polarizers are mostly effective when the angle to the sun is well off to one side of the lens (above/below/left/right). If the sun is nearly directly ahead of you or nearly directly behind you a polarizer wont be effective.