I had an idea that one kind of prison a tetronian might use would be one were the prisoners could move freely through two of the horizontal dimensions but were there might be very little room to move in the third horizontal dimension for instance the walls facing ana and kata might be the same distance apart as the width of a prisoners arm span. Like prisons in 3d I would imagine that they would also make use of the up dimension in prisons with prisoners having cells in multiple flores. I was also thinking that a tetronian prison might be a sphere.
This could have the advantage of causing a tetronian prisoner to forget that he/she is in prison and therefor not think about escaping assuming the prisoner would forget that two of its directions to move in were restricted as it would never reach any edges when moving in the unrestricted directions.