Durrr, you spin the cup, not the water. Since the water has inertia, it doesn't spin with the cup. Spinning the cup results in breaking the adhesion between the water molecules and the cup. Lift the cup quickly enough and surface tension holds the entire thing together, although it won't keep its form forever.
Seriously, haven't you guys ever seen the slo-motion videos of water balloons popping?
http://www.youtube.com/watch?v=JrH5j5kHC28#t=0m52s
You can see that the water keeps its shape as a water balloon as it falls through the air. When it pops, the rapid movement of the latex breaks the adhesive bonds without disrupting the surface tension of the water.
I tried this in my kitchen with the cup. It's really cool when you get it to work. It took me 3 tries before I was successful, so don't give up too quickly! Give it a try!