Etiquette has always been an important part of warrior behavior. Eastern warriors carried their swords and drew them from the left side, so they would place their swords on their right side as a symbol of their peaceful intent. This placement made it difficult to draw the sword quickly, thus rendering it ineffective. Failure to complete this customary behavior while in the presence of a superior was a grave breach of etiquette. Western medieval knights also had a strict code of behavior. When greeting each other, they took hold of each other's right hand, the weapon hand, and kept hold while interacting at a close distance.
For centuries, warriors have had strict protocols, such as how to entered rooms, where to sit in a room when superiors or other warriors are present, and how to remove, wear, or handle weapons in the presence of others. These protocols are strictly observed so that a warrior does not appear threatening. Modern warriors have the same protocols. Some say the military salute protocol shows that one's weapon hand is empty and therefore poses no threat.
In the military, officers do not salute enlisted personnel or junior officers, they only return a salute rendered to them. In general, senior ranks do not bow to junior ranks, they only return bows rendered to them. The person in a junior position renders respect to the person in a senior position and the senior person returns the gesture, not vice versa. The President of the United Sates has always been considered the most powerful position in the world. As such, the President on returns salute or bows, he or she does not initiate them when greeting Far Eastern dignitaries and he or she does not bow lower than the dignitary since it expresses subjection.