Helmets and Mouthpiece Retaining Straps

Nothing stirs up cave divers like discussing helmets! For some, there are the devil's work and for others, they wouldn't leave home without one. I'm a helmet guy and I present the case for and against, and show some different styles which suit cave diving, dry caving or both.
Secondly, the important topic of Mouthpiece Retaining Straps ('gag' straps) and why if you dive rebreathers they might just save your life. But can these straps and helmets get on together?