Shaving the right side of the head before the left
Anas (radi Allahu anhu) reported: When Messenger of Allah (sal Allahu alaihi wa sallam) went to Mina, he came to Jamrat-ul-Aqabah and threw pebbles at it. After that, he went to his lodge in Mina and sacrificed. Then he called for a barber and pointed his right side to him, said, “Shave from here.” Then he pointed his left side and said, “Take (hair) from here.” Then he distributed his hair among the people.
[Sahih Al-Bukhari and Muslim]
Another narration states that after the Messenger of Allah (sal Allahu alaihi wa sallam) had thrown pebbles at Jamrah and sacrificed an animal, he turned the right side of his head towards the barber who shaved it for him. Then he called Abu Talhah Ansari (radi Allahu anhu) and gave his hair to him. Then he turned his head to the left side and asked the barber to shave it. He (sal Allahu alaihi wa sallam) gave the hair to Abu Talhah (radi Allahu anhu) and told him, “Distribute it among the people.”
After returning from Arafat on the 10th of Dhul-Hijjah, pilgrims throw pebbles at Jamrat-ul-Aqabah. This is followed by the sacrifice of an animal, the shaving of the head and the farewell Tawaf (circumambulation) around the Kaba. All this is to be observed on the Sacrificial Day in the exact order of priorities. Yet, an unexpected disturbance in this order involves no juristic disadvantage. Moreover, the act of throwing pebbles marks an end to the ban on legal concessions. But this permissibility is minor and partial, that is, everything becomes lawful except the physical contact with one’s wife. The ban is totally lifted in the wake of the farewell circumambulation of the House of Allah.
Secondly, we are told that the ritual shaving of the head should begin from the right side.
Thirdly, there is a mention of the distribution of the Prophet’s (sal Allahu alaihi wa sallam) hair as a blessing among the Companions which is one of the Prophetic characteristics.
And Allah knows best!