The direct views of the objects and their mirror images generate quite different interpretations of the 3D shapes. They look like vertical cylinders, but their sections appear to be different; in one view they appear to be rectangles, while in the other view they appear to be circles. We cannot correct our interpretations although we logically know that they come from the same objects. Even if the object is rotated in front of a viewer, it is difficult to understand the true shape of the object, and thus the illusion does not disappear.
Update and spoiler alert: Via Reddit, Make Anything tries to figure out the cylinder illusion by reverse engineering with a 3D printed shape.
2018 Update: Behold another cylinder illusion example, uploaded by Hiroshi Uchinaka.
3D printed versions of Sugihara’s design are available at Shapeways.