I followed JJonathan Lampel instruction in edge modelling cartoon head, pt 5.
I selected, hit Control M on the Y-axis, and the flipped it weird ( see the pic below) at the end I had to selected it and pull them in to the effect on the second picture, and for same reason it loops weird to me, does it?
Your eyeballs are probably rotated, maybe something like this:
so that mirroring along the Y-axis doesn't work as you would like.
If you have rotated the Sphere in Object Mode, you can fix it by pressing ALT+R to re-set the rotation and then rotate 90° around the X-axis again, but if you accidentally rotated in Edit Mode, it is probably the best to delete the eyeball and start with a new UV-Sphere.