We don't expect insights to solve all our math problems. Let's do some reasoning. First, we want to avoid the shifted list. This can be done by always having 1 in the first position. Now we want to avoid reversal.
Reversal is a directional thing.
We have two directions, or more correctly, two senses. It helps to imagine the numbers arranged around a circle. Notice that if we control two slots in the list, and let the rest run free, we may still get a reversed list. If we control three consecutive positions, then reversal cannot come up again. For example, control the first three positions in a list.