Ordering sets of integers based on their elements: can we always achieve the perfect balance? Consider a family $\{S_i\}_{i=1}^n$ of finite, pairwise disjoint sets of positive integers, i.e., $S_i\subset \mathbb{N}$ and $S_i\cap S_j=\emptyset$, for any $1\le i,j \le n$ and $i\neq j$. Denote by $\succ$ a linear order over $\{S_i\}_{i=1}^n$ and, if $\overline{s} = (s_1, \dots, s_n)$ is a tuple with $s_i\in S_i$, say that $\overline{s}$ induces $\succ$ if it holds that $s_i > s_j$ if and only if $S_i \succ S_j$.
Example ($n=2$).
If $S_1 = \{0,3\}$ and $S_2 = \{1,2\}$, the tuple $(3,1)$ induces the order $S_1\succ S_2$,
whereas the tuple $(0,1)$ induces the order $S_2 \succ S_1$. Note that different tuples of integers can induce the same ordering, e.g., the tuples $(3,1)$ and $(3,2)$ both induce the order $S_1\succ S_2$.
Intuitively, I am drawing a number $s_i$ from each $S_i$ and using the
ordered sequence thus obtained to determine the ordering over the $S_i$'s.
The question I am interested in is whether, for any $n\geq 2$, we can choose the sets $\{S_i\}_{i=1}^n$ such that every possible linear order over $\{S_i\}_{i=1}^n$ is induced equally often.
Example ($n=2$, continued).
With $S_1$ and $S_2$ as above, there are two possible linear orders over $\{S_1, S_2\}$, i.e., $S_1 \succ S_2$ and $S_2\succ S_1$, and four possible tuples,
i.e., $(0,1)$, $(0,2)$, $(3,1)$, $(3,2)$. Note that $(0,1)$, $(0,2)$ induce $S_2\succ S_1$ and $(3,1)$, $(3,2)$ induce $S_1 \succ S_2$, so in this case each of the two possible linear orders over $\{S_1,S_2\}$ appears exactly twice.
Example ($n=3$).
The following assignment works:
$S_1 = \{2, 5, 7, 12, 15, 16\}$,
$S_2 = \{1, 6, 8, 11, 14, 17\}$,
and $S_3 = \{3, 4, 9, 10, 13, 18\}$.
There are $3!=6$ linear orders over $\{S_1,S_2,S_3\}$,
the sets $S_1$, $S_2$ and $S_3$ each have $6$ elements,
there are $6^3=216$ triples consisting of one element from each,
and each linear order over $\{S_1,S_2,S_3\}$ is induced exactly $\frac{216}{6}=36$ times by these triples.
The example for $n=3$ was found with the help of a computer,
and it is not unique,
but for $n \ge 4$ the search space already gets too big to handle
and the code timed out before finding any solution. Of course, the $S_i$'s do not need to have the same cardinality.
I am not necessarily looking for a solution,
but would appreciate any insight that could help me think about the general case.
Has the problem been studied before?
Does it ring any bells?
 A: I believe your question is answered affirmatively by Wen Chean Teh in “Parikh-friendly permutations and
uniformly Parikh-friendly words,” in the Australasian Journal of Combinatorics. link
In order to see this, it helps to recast your question this way:
Given an alphabet $\{a,b,\dots, x\}$ of $n$ letters, does there exist a word $w$ using all of those letters so that among the ways of underlining one of each letter within the word, every permutation of the $n$ letters appears equally often? Alternatively, does there exist a word $w$ for which each permutation of $ab\dots x$ appears equally often as a subword of $w$?
Such words correspond to solutions to your question via the following correspondence. A word $w$ over $\{a,b,\dots, x\}$ determines $n$ sets of integers $S_a\dots S_x$, where $S_\ell$ is the set of positions where the letter $\ell$ appears in $w$.
For $n=2$, and alphabet $\{a,b\}$, the word corresponding to your solution is $abba$. For $n=3$, $w=baccababccbacbaabc$. (Your sets are the positions of the letters $a$, $b$, and $c$ in $w$.)
Teh enumerates the $66$ minimal ($|w|=18$) solutions for the case $n=3$ in the paper’s appendix. Teh further gives solutions for $n=4$ with $|w|=96$. Specifically, the concatenation of these four words (in any order) is a solution:
$w_1=abcddcba\ dbaccabd\ cbaddabc\\
w_2=dcabbacd\ acbddbca\ bcdaadcb\\
w_3=cdbaabdc\ bdcaacdb\ adbccbda\\
w_4=badccdab\ cadbbdac\ dacbbcad$
Finally, Teh gives a constructive proof that solutions exist for all $n$, but the sets of integers they would determine are of size ${\left(n!\right)^{n-1}}$. It seems likely to me (from Teh’s paper and comments below) there might be solutions with sets of size $n!$.

Remarks from an earlier version of my answer, before discovering that the problem was solved by Teh.
First, a conjecture: If $\mathcal S\{S_i\}_{i=1}^n$ has the property you describe, must it be the case that every proper subcollection of $\mathcal S$ has the same property?
At a glance, I think each of the three pairs $\{S_i,S_j\}$ in your example for $n=3$ induces each of the two orders $S_i\succ S_j$ and  $S_j\succ S_i$ equally often. If this is true in general, perhaps it helps to look for a way to locate elements of a new set $S_{n+1}$ relative to a solution to a smaller problem.
Given $abba$, there appears to be no way to place $c$’s to get a solution for $n=3$. (I didn’t prove this, but I assume you searched for smaller solutions than the one you found, and it just seemed impossible...)
However, the idea of building a solution from a previous one isn’t necessarily a dead end. In your solution for $n=3$, If you look at your pairs $\{S_i,S_j\}$ in your solution for $n=3$, the words describing those pairs as solutions for $n=2$ are (with spaces added to highlight an observation) either $abba\ abba\ baab$ or $baab\ abba\ baab$, that is, repetitions of shorter $n=2$ solution words. The locations of the $c$’s has some nice symmetry: $ba\color{red}c\color{red}cab\ ab\color{red}c\color{red}cba\ \color{red}cbaab\color{red}c$
Naively, since your example for $n=2$ uses $2!$ copies of each letter $a$ and $b$, and your example for $n=3$ uses $3!$ copies of each letter, a solution for $n=4$ might exist with $24=4!$ of each letter.
While possibly still intractable, that suggests looking for an $n=4$ solution by starting with four repetitions of $baccababccbacbaabc$ (possibly with some permutations of the alphabet) and then trying to insert $24$ $d$’s to make things work ($6$ of them within or next to each of the four repetitions), perhaps with some symmetry.
