This is pretty interesting, follow the link and watch the motion of the blue lines, then turn on the occluders..
Technically the mechanism at work here is known as “motion binding”. When the edges of the diamond are covered by occluders with the same colour as the background (here, white) there is no information on the vertices of the square. Now the ends of the line become a property of the line and there is insufficient information to detect the circular movement.
The reappearance of the diamond (with occluders invisible) on eccentric viewing is probably caused by blurring the distracting line ends.