Hi Anna and Lisa
Yes it really defies logic and commonsense. It’s interesting that you can go to one yoga class and be told that you create a traction with your feet against the mat in a warrior pose, only to go to another class and get told you have to FIGHT the slippery carpet to prove that you have strong inner thighs.
It’s very dogmatic I know and takes your own intuition out of the equation (especially if you feel that potential harm is imminent). And then here’s a question to ponder: What would happen if you didn’t turn your towel, said “thanks for the suggestion, I am working toward that goal.”
As you will have read in the other posts ... as hard as it is to NOT be intimidated, it is your stuff if you are. It’s confronting and could even be abusive at times. But it’s also ultimately your choice to do as you deem fit for YOUR body. It’s only a towel. And I agree, stepping out on slippery sweaty carpet is unhygienic and damn icky. I know that in some studios I have had minor foot infections from the micro scratches in my skin and then infecting them with that less than optimal environment.
You can either stand your ground - EQUANIMOUSLY (no reaction and happy in your choice) or bow to the pressure. It’s a balance. Don’t be pushed around and don’t let them steal your peace. Your reaction will be what fuels further reaction. It’s a test! 
Namaste
Gabrielle