Rotations Are Not Single Round Here

If you haven't seen the problem 'A Roll of Patterned Paper' you may want to try that first.

In that problem I'd been finding designs for streamers - this problem follows on from that.

I noticed this about the streamers that had rotation symmetry :

In the ones I tried, if there was one centre of rotation there always seems to be another centre that worked as well.

Your challenge :

Can you find a design with only one centre of rotation?

Or if you thought that was in fact impossible, could you say why that's true ?

Taking it further :

I noticed that if there are mirror lines across the strip, and if there is also a horizontal mirror line, then the strip always seems to have rotation symmetry.

Can that be explained?