This may not apply to your situation, but when I flip my chair over, I have a clip which holds the rest of the chair to the base. It is located in the cylinder at the center of the star which attached to the pneumatic stuff.
If I pull that off, it enables me to pull the chair off of the base. I performed the same operation on my chair recently and it worked great. I did notice that when spinning in the chair it started creaking a bit. A little WD-40 over and around the clip took care of the problem.
I think I jimmied it off with a pair of needle nose pliers and a flat head. I was super careful although it may not have mattered as it appeared to be a pretty sturdy metal.