First of all the custom of saluting has evolved gradually to come to be what it is today. It is believed that it began in the late Roman time when a citizen who wanted to meet a public official had to raise his hand in order to show that he had no concealed weapon in his palm. This however grew into a means to show respect.
The army’s salute is conducted with the palms open, facing the person being saluted to show that they are not carrying any weapons and that they can trust them.
In the olden days, since the sailors were always working on their ships, their hands would get greasy and dirty. So they started saluting with their palms facing down, so as to not disrespect their seniors.
Indian Air Force
The Air Force salutes with their palms at 45 degrees to the ground showing progress towards the sky. Earlier it used to be like the Army’s salute with the palm facing up.