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.