Thank you, Bob.
You have confirmed what I suppose I already suspected. People did reverse their secular given and family names. But it's great to hear some concrete examples.
I once knew a guy in the UK with the surname MORRIS. His grandfather or great-father appeared in secular records as Breslauer MORRIS. That was a pretty obvious case of name reversal. It did not take long to find the record of the adoption of the surname BRESLAUER by a man called Moritz.