I say, and with Allah's guidance: Rinsing the mouth and inhaling water through the nose with the right hand in ablution is a matter of etiquette. It was narrated from Aisha, may Allah be pleased with her: "The right hand of the Messenger of Allah (peace be upon him) was for purification and food, and his left hand was for relieving himself and for anything that was impure." This is found in Sunan Abu Dawood 1/55, Al-Shu'ab Al-Iman 5/77, and Al-Sunan Al-Kubra by Al-Bayhaqi 1/113. Refer to: Tabyin Al-Haqaiq 1: 6-7, Majma' Al-Anhar 1: 16, and Bada'i Al-Sanai' 1: 23-24, and Allah knows best.