The design uses the thermal inefficiency of solenoids as a method to counteract working gas cooling. Working in concert, the engine would theoretically enjoy better efficiency that either a purely pneumatic or solenoid engine would on its own. That being said, it is likely the engine could run (poorly) off of either power source in a pinch.
The electromagnetic nature of the solenoids allows for in-operation tuning of timing and pull, allowing some fine control.
In fact, when run in reverse and with different solenoid timings, the engine can work as a fairly high pressure ratio compressor. The adiabatic efficiency wouldn't be great, but the system could theoretically recharge an air reservoir.