I see quite a few showjumping horses that use shadow roll(er)s and of course they are quite common in harness racing.

What is their purpose? Is it to keep the horses head up or down?

I can't imagine it would be down in a harness racer, maybe it's so they can't see the shadows on the ground, or in the case of a jumper the ground line (not sure why you would want them not to see it though!)

I think in harness racing it is used as a tool to help aversion to shadows and in SJ to keep their head down especially if they don't like martingales or throw their heads up and lose their concentration.

See, we have the same two different opinions that I had so I'm still bamboozled!!! :-)