Why $H$'s radial component in a solenoid filled with linear magnetic material is zero?

Why $H$‘s radial component in a solenoid filled with linear magnetic material is zero?

Unlike $B$ where we can use the Biot-Savart law to say that reversing the current results in reversing the field, this argument cannot be used on $H$ and therefore needs a justification which Griffiths does not provide.