Kokichi Sugihara created the video "Ambiguous Optical Illusion: Rectangles and Circles." In it he shows a 3-dimensional shape that looks like a circular cylinder but in a mirror looks like a square cylinder. The following example shows how he achieved the effect.
The closed curve in the applet below represents the top rim of the shape. If you view it with the positive green and blue axes lined up, it will look like you are viewing the unit circle in the green-red plane. To do this, arrange the axes so one red dot sits atop the other red dot. If you drag the image so that the positive blue axis lines up with the negative green axis, it will look like you are viewing a square (oriented as a diamond) in the green-red plane. To do this, align the two blue dots.