We'll look first at this 3-legged shape, based on the symbols of the Isle of Man and of Sicily.

This shape fits exactly onto itself three times when it is rotated through a full turn about its centre. We get an unchanged appearance when the tips of the toes (or any other three similar reference points) have moved through one third of a turn, or two thirds of a turn, or a full turn so that they have come back to where they started, giving the same effect as staying put. Doing any number of repetitions of these three turns will not take the toes to anywhere other than these three positions.
This shape has what's called rotational symmetry of order 3.

There is a new possibility for this shape. Each wing reflects exactly onto the other wing about a mirror line running through the centre of the body. On the other hand, rotating the butterfly shape so that it looks unchanged is only possible if you take it through a full turn. This means that the butterfly shape has two possible symmetry transformations.