A320 vs. B737-800 Fuel Burn

There is the never-ending "war of words" between Airbus and Boeing of which aircraft is the more fuel efficient one, and, since the launch of the A320neo, if the A320neo would be more fuel efficient and if yes, by what margin. Further complicating, Boeing meanwhile claims, that it's yet-to-be-named reengined B737 would have the same advantage in terms of fuel efficiency as the current generations of these two narrowbodies have. And as even Boeing does not seem to know how the exact configuration of the reengined B737 looks like, this claim seems (at least) a little bit premature.
As I tried to lay out in an earlier posting, much depends on what you want to compare:  fuel burn per trip of fuel burn per passenger.
The official documents provided by the manufacturers