Three Organizations

Robert Heinlein's Temporal Bureau operates in an immutable timeline. Temporal agents occasionally close causal circles.

James Blish's Service operates in what its members hope is an immutable timeline although they take the precaution of causing future events that they know about, thus generating causal circles.

Poul Anderson's Time Patrol operates in a mutable timeline. Patrol agents prevent causality violations, often by completing causal circles.

Of these three scenarios, only the Time Patrol became a series because, in an immutable timeline, a circle, once closed, is complete whereas, in a mutable timeline, a further causality violation or quantum fluctuation remains possible. However, the Service receives messages from futures described in two otherwise unrelated stories, thus generating a very loose series or sequence, which might have gone on to include Blish's finite spinning universe version of time travel. One of the messages received is from a world-line cruiser traveling backwards in time and another is from a future era that has received a time projection from the twentieth century.

By contrast with these works, SM Stirling's Nantucket Trilogy focuses on the detail of a single population transported to the past of an alternative timeline.