If $S$ is a set and $G$ is a group of functions from $S$ to $S$ under composition (not necessarily all functions $S\to S$, but closed under composition), and additionally for all $s_1, s_2$ in $S$ there exists a unique $g$ in $G$ such that $g(s_1) = s_2$ does this have a technical name?
As a concrete example, $S$ could be the set of all date times and $G$ could be the set of all functions that add / subtract a fixed duration from those date times
The name is not timeline algebra :)
In general groups which come from functions like that are called permutation groups. An important, although easy, result in group theory is that all groups can be represented as permutation groups.
The restriction you give are restrictions on the natural group action on $S$. In particularly I would describe your type of action as a free and transitive action. I have also heard simply transitive or sharply transitive to describe this.
A free action describes the uniqueness, and transitive is the for any two points $x,y$ there is a $g$ such that $g(x)=y$ part. Every group has a free and transitive action on a set by acting on itself by left multiplication. It should be clear that not all groups behave like your timeline algebra example, groups can be way more complicated!