I have been studying they symmetries of the square, D₄. I would like to use the subgroups that are not normal to make a coset space, but since they are not normal, the right coset space is different than the left coset space. However, I did notice that the left and right coset space of pairs of subgroups in the same conjugacy classes overlap. Is there a name for this whole system?
This picture is one of the two conjugacy systems. There are four overlapping coset spaces in the form of the two rows and columns.
Contrary to the comment, these are double cosets. They are not all of the double cosets. They are simply the double cosets that are also cosets. Usually double cosets, being unions of cosets, are larger than cosets, but when the a left coset $H_1g$ of $H_1$ coincides with a right coset $gH_2$ of $H_2$, then $H_1gH_2$ coincides with both of them. This happens precisely when $H_1$ and $H_2$ are conjugate, and the number of such cosets is $[G:H_1]/|Orb| = [N_G(H_1):H_1] = [N_G(H_2):H_2]$, where $|Orb|$ is size of the orbit of $H_1$ under conjugation.
In this case $[N_G(H_1):H_1]=2$ since $N_G(H_1)$ is generated by $H_1$ and the center (the 180 degree rotation), so there are two coset-sized double cosets in $H_1\backslash G/H_2$ and similarly for $H_2\backslash G/H_!$. In each case the other double coset has the other $4$ elements of $G$, for a total of $3$ double cosets in each case.
Specifically, if I am able to read your picture properly (the fonts in it are a bit blurry), $H_1\backslash G/H_2$ consists of the double cosets $H_1eH_2=\{e,rf,rr,rrrf\}$, $H_1rH_2=\{r,rrf\}$, and $H_1fH_2=\{f,rrr\}$. Note that the double cosets with two elements are precisely the sets that are simultaneously left cosets of $H_2$ and right cosets of $H_1$.
I'll leave $H_2\backslash G/H_1$ as an exercise so you can validate your understanding.