Temple
Generally a bureaucracy in most settlements, though the details vary wide. Commonly has several aspects, often called Orders. The orders of the temple will often have a local name in a region, but are collectively referred to as "the temple". This often annoys the politically minded members of various Orders as they are not usually associated with each other in any formal manner. That said, they often cooperate given then tend to focus on the proper functioning of society in a strange world.
See also Temple Sorcerer
Deals with funereal rites, avoiding creating undead, and dealing with undead that arise. Most folks are generally intimidated by these folks as they have their death association, but also because they are one of the classically armed orders of the Temple -- after all, they deal with undead problems. Clerics are usually of the Grave Domain, but can be of others.
Imagine the folk from Katherine Addison's works who deal with the dead, minus the criminal investigation angle. Or, imagine the Order of the Bastard from Bujold -- except every untimely death has a good chance of spawning a demon.
Seeks to understand how and why the world works as it does, particularly where it seems that causality breaks down. Fundamentally cosmographers. There is a distinct division in the Order of Mysteries between those who theorize on the why things work the way they do, and those who gather data (referred to as "givens") in the world about them. The theoretical side are referred to as Theoreticists of Theors, the agents who gather givens as Investigators.
Imagine the mathic folk Anathem, except without the Consents, and add in the romantic version of Victorian explorers or Indiana Jones crossed with the X-Files for the Investigators who gather givens for the Theors. A given Investigator is generally associated with a particular Theor or group of Theors, and investigates phenomena of interest to them.