This effect was noticed while reading the tutorial page for the rotate() function in Processing. As the square rotates it appears to have localized variations in brightness which are anchored to its shape. Below is an image attempting to reproduce the perceived image:

Specifically, there is a darker ‘X’ shape going towards the corners which leaves brighter inclusions at the centers of each side. The very tips of the corners also appear bright. This luminance pattern is reversed for a black square on a white background, and diminishes as equiluminance is approached. It seems like at lower rotation speeds, the dark area shrinks to a circle filling the center of the square. This could be habituation to the brightness of the parts of the screen that are always on, but the four-leaf-clover-esque pattern visible at higher rotation speeds appears to be as yet unexplained.

Below is an interactive demo:
-Mouse X controls rotation speed
-Mouse Y controls square size
-Clicking controls the direction of rotation
-UP/DOWN keys alter the luminance of the square
-LEFT/RIGHT keys alter the luminance of the background

Two further demos were produced to explore this effect in translating and flickering squares:

Translating square
-Mouse X controls translation speed
-Mouse Y controls square size
-Clicking controls the direction of translation
-UP/DOWN keys alter the luminance of the square
-LEFT/RIGHT keys alter the luminance of the background

Flickering square
-Mouse X controls flicker frequency
-Mouse Y controls square size
-UP/DOWN keys alter the luminance of the square
-LEFT/RIGHT keys alter the luminance of the background