I recently moved into a small neighborhood in Tucson, and I noticed a behavior that I evolved spontaneously: when I'm out and about I wave or say hello to everyone driving or walking by within the bounds of the neighborhood. I do that even though I usually tend to be reserved and keep to myself. It seems like a simple gesture, but it shows members of the community that I'd like to belong and be on good terms with the people in these 40 properties. Most people in a new neighborhood would probably do the same. This is a modern form of reiho or etiquette.
There are lots of reasons for reiho in Japanese culture that go back many centuries. In about 1380, Ogasawara Nagahide codified nine different kinds of kneeling bows for different situations, for instance. If you want to read about this history, Dave Lowry's book In the Dojo is a great place to start.
In the modern dojo, I tell my students that reiho is like a not-so-secret handshake that signals good intentions within a communal space. At home, waving to my neighbors establishes friendly intentions and signals participation in the norms of the larger community. When we bow to each other on the mats, we signal that we are a part of the training community and that we intend to protect our partners by following the shared rules of practice established by the teacher. Each time we bow, it is both a reminder and a re-affirmation of this agreement, me to you and you to me.
We also see people bow when they enter and leave the dojo and when they come onto or leave the floor. The usual rationale for this is that we are showing respect to the art and the place of practice. This is true. However, it is also true that we are pausing to remind ourselves that ours is a serious and dangerous practice. It is a time to gather our wits and bring our attention to bear as we cross the threshold. In the dojo where my karate group trains, the class immediately preceding mine is a Japanese sword class. I make a point (pun totally intended) of making sure no one is swinging a sword near me when I step onto the mats. Stopping for a moment to bow reminds me to pay attention.
Reiho is about respect, and respect leads to safety. It is how we create a civil culture in general and create a dojo where people support communal well-being rather than seeking to profit at the expense of the health of others. So wave to your neighbors, and bow to your partners. Everyone wins.
