So I should probably explain the thinking behind this rather than just clown on myself. This design is a dual-expansion air engine, integrating in a stroke-offset shuttle piston that acts as an intermediary between the high and low pressure volumes.
Every stroke is a power stroke, and the solenoids help extend that power stroke to include the half of the upstroke as well.
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.