If $H$ is a subgroup of a group $G$, is there a standard name for subsets of the form $xHy$?
As @user1729 said in comments, these are conjugates of cosets. But they can also be described as left or right cosets of subgroups conjugate to $H$.
So $xHy = xy(y^{-1}Hy)$ is a left coset of the subgroup $y^{-1}Hy$.
And $xHy = (xHx^{-1})xy$ is a right coset of the subgroup $xHx^{-1}$.
I remember a while back someone asked for help with an exercise that said "prove that every left coset in a group is a right coset", and several people replied that this was wrong, but in fact it is correct because $xH = (xHx^{-1})x$.