A few years ago I studied pentacube oddities. A pentacube is a polycube with 5 cells, and an oddity is an arrangement of an odd number of copies of a polyform that has binary symmetry (or stronger). So far as I know, Torsten Sillke was the first to study oddities in general, and polycube oddities in particular. Mike Reid has also studied them.
In particular, I wanted oddities with full (cubic) symmetry. This page summarizes my results: https://userpages.monmouth.com/~colonel/c5odd/index.html.
This catalogue has weaknesses. Among achiral (mirror-symmetric) pentacubes, my solutions for the T, W, and Z pentacubes are made from rectangular boxes known to be tilable. They have hundreds or thousands of tiles. I have no solution for the X pentacube.
Among chiral pentacubes, my solutions for the H and S pentacubes are formed from rectangular boxes. They have 145 and 675 tiles respectively. I have no solution for the G pentacube unless we let the tiles be reflected.
Can anyone improve on these results, or suggest a method for improving them?