PUZZLeaTOMictM

Intro    Part1     Part 2    Part 3  Part 4

Note that these 5 rules could be made with less rules since some of them imply the others.  Also a twisted structure in three dimensions can have 3 axes of symmetry.  Think of a double cylindrical spring.  It has standard x,y,z axes of 180 degree rotational symmetry.  So you could easily make a symmetrical polyhedron with a twist with the same three axes of symmetry.