I won't replay the working principle of an Alpha-Stirling here (for the new readers), and this discussion will assume the interested reader is already familiar with how they work.
If we apply heat and cold to an Alpha-Stirling, the crankshaft will be driven with power. If we reverse that and a motor is attached to the crankshaft, spinning the Alpha with power results in one side getting warmer and the other side getting cooler. We can see this principle at work from the many cryo-coolers which throw the heat away, and use to cold side to liquify oxygen and nitrogen (among other profitable super-cooling jobs).
The link I will add below, along with the picture I will also post here, is a driven Alpha, with the hot side and cold side heating one loop of silicone oil, and cooling another loop of silicone oil. I suppose in the summer you would "throw away" the hot air, and use the cold air to make your home more comfortable, and...in the winter you would throw away the cold air, and use the hot.
There are two interesting things about this for me. I've seen several clever proposals on youtube for scaling up some kind of "fluidyne" Stirling. I found those when researching liquid pistons in a "U-pipe". As I'm sure many of you know better than me, the thought of using helium as the working gas adds several complications, the main one being how to seal the pistons. Bellows have benefits and drawbacks. We "could" encapsulate the engine except for the outsides of the hot and cold ends. Pressure inside the cylinders would rise and fall, but the crankcase would contain a "medium" pressure of helium, which should balance out any piston leakage. Who knows?
Anyways, there were several interesting features that might have some application to a potential design bouncing around this forum, so I thought I'd post it. The link starts the youtube at 6:55, where the animations are. The "R" is the regenerator. Low restriction to gas flow, lots of area to transfer heat/cold.
https://youtu.be/X1fiABe4x08?t=415
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The second thing that really caught my eye was the finned pistons. He mentions that a key issue is using a Stirling engine/pump is heat transfer. Of course you can switch to helium to improve that a little, but even with helium, can we improve that? When a piston is moving through an open cylinder there can be quite a lot of stratified flow. A thin layer of gasses that are touching the cylinder walls is heated or cooled, but a large portion of gasses in the center do not absorb/shed much.
Imagine taking a very long piston that is almost solid aluminum/copper, and on the crown of the piston (the part facing the gasses), you cut very deep grooves, so deep that 90% of the piston volume becomes fins. These heat-pump pistons have the crank above vertical cylinders, and the pistons are driven in a downwards direction into the silicone oil below. The gas is squeezed or expanded when the piston goes into the oil, or is pulled up.
For our purposes, below each piston could be a similar set of fins that mesh with the piston (rather than silicone oil). These could be made from stacked plates (perhaps aluminum on the cold side and steel on the hot side?), and they could be made in such a way that they form a cylinder in its center. In this way, the cylinders could have fins on the inside and outside of the cylinders, formed by stacked plates, cut to shape? I'll post a drawing later to clarify.
The video from the company that is making these suggested that two millimeters was the optimum gap for heat transfer from closely-mounted fins to a working gas.