I understand that there's sort of a reflector at the back of a cat's eyes that magnifys light and helps cats to see in low light situations. It works something like a mirror or glass of any sort and reflects light back at the camera when it hits dead on.
Now light will bounce off something like that at a reflection angle that is the opposite of the one the light is coming from, (the law being known as the angle of reflection is opposite to the angle of incidence - the angle it's coming from). The only way to avoid it is not to have the light bounce back to the camera, which means you can't take a direct on flash photo, you must take it from an angle. so that the light will bounce away from you. You can experiment by taking pics of reflections in a mirror.