Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry (permutation symmetry) ?

The motivation comes from physics, with a geometric example:

Take one $D_2$ disk, with n indexed "intervals" "punctures" ($I_1, I_2, .... I_n$) living on the boundary $S_1$ of the $D_2$ disk. There is a $Z_n$ symmetry, but not a $S_n$ symmetry, because the "intervals" "punctures" cannot cross each over (there is only one dimension on $S_1$)

Take an other identical disk, with n indexed "intervals" "punctures" ($I'_1, I'_2, .... I'_n$)

Now, we are gluing the two structures. We deform the two disks as half-spheres, and gluing them at the equator, to obtain a 2-Sphere. We deform each interval $I_i$ or $I'_i$ as a litte half-circle, and gluing, at the equator, each couple of indexed "intervals" "punctures" ($I_i,I'_i$"), so that we obtain n indexed little disk boundaries $D_1, D_2, ......D_n$, at the surface of the sphere.

But now, we have a permutation $S_n$ symmetry, because, roughly speaking, we can freely move the little boudaries $D_1, D_2, ......D_n$ at the surface of the sphere , and so exchange their place.