The first rule of puppet pin: The grid is created on the very first frame of the layer using the edges of the alpha channel of the layer. This grid is fixed and any movement layer will not move the grid.
What that means is that you cannot rotate, scale or change the position of the layer if you are using puppet pin.
If you want to rotate the petals of your flower but keep the center fixed the center you need pins that don't move in the center. You just do not have enough pins on the layer.
It is possible to tie pin positions to a Null using expressions and then rotate the null to move a group of pins at the same time. Here's how I would set that up:

Here's the workflow:
- Add six pins to the outside center of each of the petals
- Add six more pins to the inside of the petals to hold them in position
- Add a null and name it Null 1 Rotation
- Duplicate the null and name the duplicates Null 1 follow 1 through 6 then parent them all to the rotation null
- Copy the position of the first six puppet pins to the position of the six corresponding follow nulls
- Add this expression to Pin 6
thisComp.layer("Null 1 follow 6").toComp([0,0])
- Select the Pin 6 property and copy expression only
- Select Pins 1 through 5 and paste
- Edit the expressions in 1 through 5 so they point to the proper nulls
- Animate rotation
Ther you go. I added a loop out Expression to the rotation property to make the flower wind to the right and then to the left and repeat.
If you are interested here's the project file: https://www.dropbox.com/s/raktbnkfo5wnubo/puppet%20pin%20to%20null.aep?dl=0
(note: If your browser adds a .txt extension to the AEP file just delete it.)