[tStano]

I believe the physics of this doesn't work. As I understand it, you're lifting with the difference between venting pressure and pressure in the system. In the usual system, your venting pressure is 0(gauge pressure), and working pressure is 60. If you were to put your 60 working pressure back into your 120 storage pressure, your cylinder would be forced the other direction, and if you put a one way valve on it, the exhaust air wouldn't be able to push into the 120 psi air.

You can also think about it in terms of the second law of thermodynamics, roughly paraphrased: "you can't make a perpetual motion machine".
All the energy in your pneumatics system comes from your compressor. If you somehow manage to use your air that has already done work to do the same amount of work again(which seems to be what you're describing), you've created a perpetual motion machine, and should be a very rich man now.