If you reversed the connections to the secondary the circuit would probably work just as well as if one side of the secondary was grounded as you'd expect.
The reason why the circuit has a common + on the coil is probably due to the fact the author has used a car ignition coil, some car coils esp older ones have a common pri and sec connection to the + terminal, making that circuit necessary.
You could just reverse the low tension polarity, however that might cause issues with the coil internal insulation, they have coolant and all kinds of weird stuff in them.