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 &...
The formula for pricing a swaption under normal volatility is simply the Bachelier formula. It may be found in many papers (for example, Le Floc'h Fast and accurate basis point volatility), and is also on stackoverflow.
You can easily move from a payer ($C$) to a receiver ($P$) by using the put-call parity relationship:
$$ C(t) - P(t) = B(t,T) (F(t,T)-K)\,,$...