The stirling engine is reversible in a common meaning of the word, that means - apply temperature difference, get torque; apply torque, get temperature difference. Of course it won't be fully reversible in thermodynamic sense, due to its own lossy character of operation (like loose displacer that allows heat transfer from hot to cold regenerator without producing usable work.)
Stirling engine and stirling cryocooler are essentially the same device, they work on exactly the same principle - take unpowered stirling engine (with no heat/cold sources) and start rotating the axis and you will cause the two plates to obtain temperature difference.
Of course in practical uses there are serious technological differences that may make the operation impossible, but the principles are the same.
The four phases of stirling engine in "cryocooler" mode?
- Compress the air on the "hot" side by artificially pushing the work piston, providing external work. Compression heats the air. Hot air passes heat to the regenerator, losing some temperature.
- Displace the air towards the cold side, using the displacer.
- Decompress the air by pulling the work piston. Some work is provided by air pressure, more, external work is provided. Air is cooled and extracts the temperature from the regenerator, making it cooler.
- Displace decompressed air back to the hot side using the displacer.
- hot side regenerator transfer heat into gas. Gas heats, expands, pressure rises, this pushes work piston providing work, reducing pressure.
- gas is displaced to cold side regenerator.
- gas transfers heat into the cold side regenerator. As its temperature drops, pressure and volume drops, pulling the work piston in.
- Displacer moves cooled gas back to the hot side.