Easily the most commonly encountered symbol of the International Order Of Oddfellows is the three connected loops forming a chain. This is usually known as the "Chain With Three Links" or the "Triple Links". Occasionally they are also seen accompanied by the letters FLT, which stand for Friendship, Love and Truth. Both examples seen here are taken from graves in Cave Hill.
There used to be a great old enormous triple mirror at Richmond's Glyndon Hotel designed like the Oddfellows rings, but I'm not sure if it's still there now that the place has been remodeled.
The Independent Order of Odd Fellows have three levels of Lodge: the Subordinate Lodge, the Encampment, and the Patriarchs Militant. In addition, there is a private club named the Ancient Mystic Order of Samaritans.
According to Wikipedia, fellowship in the IOOF entails, among other things: