Do all combinatorially distinct fundamental polygons correspond to surfaces? 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. $a^{-1}a$?
 A: *

*Yes. 2. Yes. (I suppose that the surfaces are "the same" if they are homeomorphic). 


For 1, it is sufficient to check the definition of surface: that every point has a neigborhood homeomorphic to the disc. For interior points of the polygon, and for points on
the sides, this is evident, and for the corners this is easy.
For 2, just recall classification of all possible compact surfaces up to homeomorphism.
There is one integer invariant, the Euler characteristic, and the 2-valued invariant,
orientability.
The Euler characteristic is easily computed from your word:
if you have 2n edges of your polygon, they will give $n$ edges after gluing,
and suppose that you obtain $v$ vertices after gluing.
Then the Euler characteristic is $1-n+v$ which is between $2-n$ and $2$.
And orientability has two values. So you have at most $2n$ topologically different
surfaces from strings of length $2n$.
And you see that
for given length there are much
more classes of words than $2n$.
A: Great question!
There's an interesting comment at https://math.stackexchange.com/questions/204405/for-an-n-gon-how-many-fundamental-polygons-are-there
"we ... disallow discontinuities, [i.e] two A edges to be adjacent with "wrong" directions."
In the first 3 polygons of the top row, the head of A or B is attached to its own tail so these are ignored. 
Another insight is that the orientation of the arrows is a guide for attachment and is invariant if direction is inverted for all identical edges, say flip all A edges. 
So the last two polygons on the top row give the same glueing for a sphere, just flip the direction of the red lines to turn one polygon in to the other.
On the bottom row you have: Real Project Plane, Klein Bottle, Torus and Real Project Plane again (invert the red lines to equal the 1st RPP). 
