Mega-bump! But I'm hoping I can get some replies this time.

I guess I totally missed

this page the first time around - it's an enumeration of both "free" (flippable in 4D space) and "one-sided" (realistically embedded in 3D space) up to the decarhons.

I also found

this picture from the online version of Stewart Coffin's The Puzzling World of Polyhedral Dissections:

It shows all the pieces of order 2 through 4, including chiral pairs that the other page does not.

There are 28 tetrarhons in that picture, with a total volume of 112 units. Coincidentally, a rectangular "pyramid" with a 6x7 base (6 layers, 1x2 at the top) has 112 units. Can they all fit into such a shape?

(The other pieces in the picture total 17 units. They can fit into a 19-unit octahedron with two opposite corners removed (square layers of 4, 9, and 4) quite nicely - and it's simple enough that I could work it out in my head.)