the length of the night on either side of the solstice presumably would flip-flop around just as does the date of the solstice itself.
Not sure what you mean by flip-flop (too technical ;o) but taking the moment of solstice as the nadir, the length of darkness immediately before and after would be identical.
Well, as Sid Kipper points out, the longest day around here is the day the clocks go back, when there are 25 hours in it, and the shortest day is the day they go forward, when there's only 23 hours.
But that would be a non-natural day, Kevin. If you flew a jet around the globe you could have a potentially never-ending day. Like the time my Mother in Law stayed for three months.