Instead of taking all the groups and twirling them together, try dividing them into sections and laying each section over each other in one direction.
For example, you would divide the whole thing into 2 or 3 groups on a table. Then you would take the left most group and move it over the top of the other one(two). Then you would take the one that is now the left most and lay it over the others.
I drew a pic to try and see if that would help to make any sense. You aren't twisting the groups, just laying one on top of the others, but since they are all being moved in the same direction it will make it look twisted.