I leave them on as they still look good and I understand they look good with frost on too.

Mine snapped off at soil level -so I'v hung it up in the greenhouse to dry it out, never done this before so we shall see

In the past I've cut them at soil level, hung them upside down to dry and used them as decorations. But now I cut the heads off and throw them into the borders, let them roll around like tumbleweed and self seed wherever they like!

That's a good way to get healthy, vigorous new plants, but the bigger kinds will take 3-5 years to reach blooming size.