Let $X$ denote a set. There's a corresponding group obtained by taking the group freely generated by $X^2$ and then quotienting out by the following families of relations:
$(x,x) = 1$
$(x,y)(y,z) = (x,z)$
$(x,y)(y,x) = 1$
For each quadruple $(x,y,x',y')$ such that $\{x,y\} \cap \{x',y'\} = \emptyset$, we have: $$(x,y)(x',y') = (x',y')(x,y)$$
Question. What is this group called?
My motivation for considering this group is that it acts on the set $(\{0,1\}^\mathbb{N})^X,$ of $X$-many binary streams, by interpreting $(x,y)$ as the act of taking the first digit of stream $x$, removing it from $x$, and appending it to the beginning of $y$. Each of the four families above can be explained in these terms; for example $(x,x)=1$ is saying that if I take the first digit of stream $x$, and put it back on stream $x$, nothing changes.