Greetings buckybal, and welcome to the forum.
There are several terms used to describe words that can mean their own opposites, and this goes beyond just greetings. For example: oversight, dust, bolt, weather.
Don't know if any of these terms are "official," but I've heard this type of word called: auto-antonym, contranym, antagonym, and Janus word.
See this Wiki page for more: http://en.wikipedia.org/wiki/Contranym
Not being fluent in a language that normally uses one of these dual-purpose forms, I can only guess that those who are do not see them as separate and opposite usages, rather as the same formality to mark both endpoints of an interaction. If we commonly used, for example, "Peace be with you" both upon meeting and departing from another person, I don't think we would view it as antonymic, merely as a marker for "I am now starting or ending a conversation," and context would tell you which. It occurs to me that a phrase I've heard fairly often at either end of a conversation is, "Hey, nice to see you."
On second read of buckybal's question, I see I might have missed the point. Greetings that can be used at arrival or departure are probably not a subset of contranyms (my preferred term).
Ron: Just my two cents, but I wonder if good night doesn't work, and good evening does, because evening is followed by night. Evenings can involve ongoing interactions over a period of time, but night has a certain finality to it.
Tromboniator: I see your point about non-antonymity. "Peace be with you" is a great example. Not being fluent in a language, it's easy to assume that, say, aloha takes a different (opposite) meaning depending on when it is spoken, when the meanings could well be identical.
Just checked Wiki. It says aloha means affection, peace, compassion, mercy. Seems pretty time-independent to me. And all these years I thought it translated to hello and goodbye. You never know what you don't know …
Thanks! Other people at my company are based in England, so I can see "Cheers" making sense now, and though growing up in the Midwest I might have thought of "Good morning/afternoon/evening" as solely greetings, while "Good night" as just a farewell, after watching a good amount of Horatio Hornblower, "Good day" fits in well. I'd start using it, if I wouldn't sound stilted or too odd. And Tromboniator, I had noticed that wishing the other person well would be a good statement to make when seeing or leaving someone.