The reaction forces of the clothes on the drum would simply be called 'reaction forces'
Centrifugal force is a funny thing.. To say it "doesn't exist" is a little wrong.
Without too much detail; Centrifugal forces 'appear' to be real in non-inertial reference frames. For instance, imagine you are in a car, with something on the dashboard. Going around the roundabout, the something slides to the left hand side of the dash. In the reference frame of the car, some force pushed that something and did work on it, while the centripetal force was working on the whole frame. This is an example of a fictitious force, one that doesn't actually exist (since it's not observable in all reference frames) but can still do work in some. The reason why it's a little wrong to say it doesn't exist is because it turns out gravity is in fact, a fictitious force; yet we all know how real that is.