The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
the construction can be specified by a string of $2n$ symbols:
$a b a^{-1} b^{-1}$ for the torus, $a a b b$ for the Klein bottle, etc.
(*Reflecting a comment by Benjamin Steinberg*:) Each letter in a fundamental
polygon string appears exactly twice.

Let's say two symbol strings are equivalent if they are related by a
combination of (a) circular
permutation, (b) reflection/reversal, or (c) symbol permutation/relabeling.
For $n=1$, there are two distinct strings, $aa$ and $aa^{-1}$.
For $n=2$, I *think* (not certain)
these are the combinatorially distinct strings:
$$
aabb,\; aa^{-1}bb,\; a^{-1}abb,\; aa^{-1}bb^{-1},\; a^{-1}abb^{-1},\;
$$
$$
abab \;, aba^{-1}b,\; aba^{-1}b^{-1},\; a^{-1}ba^{-1}b
$$

Two questions:

Q1. Does every possible such string correspond to some surface?

Q2. Might two combinatorially distinct strings correspond to the same surface?

Perhaps $aa^{-1}bb^{-1}$ and $a^{-1}abb^{-1}$ describe the same surface, as they only differ in $aa^{-1}$ vs. $a^{-1}a$?