Why you'll never see the movement of your own eyes
The brain does not process visual information when the eyes are moving, a phenomenon called saccadic masking. This results in a brief moment of "blackness" which the brain is adept at hiding from the viewer. Thus, it is nearly impossible for one to observe the movement of one's own eyes (try it!). It is thought that saccadic masking is an adaptation which prevents confusing and useless motion blur, a potentially disorienting phenomena observed in particularly kinetic video and in certain styles of non-stabilized filming.