Materials for pictures that change with the direction of sunlight
I think what you are looking for are called “chameleon paints”, but I cannot recommend you use cardboard. It is better to suited to non-flexible surfaces.

Check out chameleon car paint or search on amazon for it. Most automotive shops should stock it.

Another option might be to look for iridescent paints, but they just get a bit shinier depending on the angle that light bounces off the surface into your eye.