Beautiful intricate math problem

Hello! Here is a really fun problem I came up with.

You are given a perfect square $n.$

Define a superpair in $n$ as a $2(n-1)$-tuple, $(x_1, x_2, x_3, ... x_{2(n-1)})$, such that $x_1\cdot x_2 \cdots x_{n-1} = x_n \cdot x_{n+1} \cdots x_{2(n-1)},$ with $1 \leq x_i \leq n$.  In other words, all elements must be between $1$ and $n$ and the product of its first $n-1$ elements equals the product of its last $n-1$ elements.)

Find the set $S$ of values of $n$ that guarantee that it is possible to form list of $n$ superpairs in $n$ such that every number that appears in your superpair list appears exactly twice in your superpair list.

For example, we know that $n=3$ is in $S$ because we can satisfy the conditions with the following superpair list:

$(1,6,2,3)$

$(2,4,1,8)$

$(3,8,6,4)$

Here, $1\cdot 6 = 2 \cdot 3,$ $2 \cdot 4 = 1 \cdot 8,$ and $3 \cdot 8 = 6 \cdot 4.$ Additionally, every number present in the the superpair list, ($1,2,3,4,6,8$) appears exactly twice.

Notice that not all numbers from $1$ to $n$ need to be in your superpair list. Rather, you must only ensure that the ones that are in your superpair list, appear exactly twice. Hint: try using exactly $n^2-n$ distinct numbers in your superpair list.

I hope you enjoy!