The SwaptionHelper class inherits from BlackCalibrationHelper class:
As a result, one of its attributes is volatilityType which can be normal or lognormal or shifted lognormal.
You can see it in the first constructor in the link you sent:
const Period &...
Hull & White is often use to value Bermudan swaptions, given a market for European swaptions. The idea is, at given mean reversion speed, to calibrate the instantaneous volatility to the set of coterminal european swaptions that correspond to each Bermudan exercise date. Hence the Bermudan swaption price becomes a function of its coterminal European ...