This is a spin (pun slightly intended) on George Hart's cube puzzle. In his puzzle, he dissected a cube into two pieces with a spiral-like surface called a helicoid. The resulting two pieces are identical, and are themselves 2-fold symmetric.
I played with this idea on some of the other platonic solids. I cut an icosahedron into two equal and symmetric pieces with a helicoid.
Going further, if you have a solid with a 3-fold axis of symmetry, you can cut it into 3 equal pieces with 3 helicoid surfaces. I did this with a dodecahedron and an octahedron. Note that these polyhedra also possess 2-fold symmetry axes, and so can be cut into 2 pieces as well.
I included the Python Jupyter notebook I used to create them. I was using SolidPython, which is essentially a Python wrapper around OpenSCAD.
In terms of printing, some of the pieces require a little support; nothing major. Note that you must print multiples of each piece; for example, if you wish to print the dodecahedron sliced along the 3-fold axis of symmetry, print 3 copies of dodec_piece_3fold.stl. If you want to print the icosahedron, sliced along its 2-fold axis of symmetry, then print 2 copies of icos_piece_2fold.stl.