Official URL: http://www.maths.tcd.ie/pub/ims/bull54/S5401.pdf

The classification up to conjugacy of the homeomorphisms of the real line onto itself is well-understood by the experts, but there does not appear to be an exposition in print. In other words, it is mathematical folklore. In this expository paper, we give a complete but concise account of the classification, in terms of a suitable topological signature concept. A topological signature is a kind of pattern of signs. We provide similar classifications for homeomorphisms that fix a given subset, and for germs of homeomorphisms at a point. For direction-reversing homeomorphisms, we show that the signature of the compositional square is antisymmetric. We go on to apply the conjugacy classification and signatures to reprove recent results of Jarczyk on the composition of involutions. His results classify the reversible homeomorphisms (the composition of two involutions), and show that each homeomorphism is the composition of at most four involutions. The reversible direction-preserving homeomorphisms have symmetric signatures.