= Solution
There are \b[exactly four homeomorphism types]: the <two-sphere>, <real projective plane>, <torus>, and <Klein bottle>.
For completeness, enumerate all pairings. Four sides have three pairings; each pair can be matched in either direction, giving twelve signed pairings before symmetries. Up to cyclic relabelling, the repeated labels are either adjacent, of form $aa^sbb^t$, or alternating, of form $aba^sb^t$, where $s,t\in\{1,-1\}$. The second adjacent pattern $abba$ rotates to the first. Endpoint identification gives the following table:
$$
\begin{array}{c|c|c|c|c}
\text{pattern}&(s,t)&V&\chi=V-1&\text{surface}\\\hline
\text{adjacent}&(-1,-1)&3&2&S^2\\
\text{adjacent}&(1,-1)\text{ or }(-1,1)&2&1&\mathbb{RP}^2\\
\text{adjacent}&(1,1)&1&0&\text{Klein bottle}\\
\text{alternating}&(-1,-1)&1&0&\text{torus}\\
\text{alternating}&(1,-1)\text{ or }(-1,1)&1&0&\text{Klein bottle}\\
\text{alternating}&(1,1)&2&1&\mathbb{RP}^2
\end{array}
$$
Here the quotient always has two edge classes and one face, so its <Euler characteristic> is $V-2+1=V-1$. To reproduce the vertex count, label consecutive square vertices $0,1,2,3$. For paired sides $i,j$, a same-direction pairing identifies $i$ with $j$ and $i+1$ with $j+1$; a reversed-direction pairing identifies $i$ with $j+1$ and $i+1$ with $j$, with indices modulo four. Taking the generated equivalence classes gives exactly the displayed values of $V$.
The quotient is <orientable> precisely when each label is traversed once positively and once negatively around the boundary: then the face orientation extends across every paired edge. A same-direction pair obstructs such an orientation. Thus the two $\chi=0$ possibilities are distinguished as the orientable <torus> and nonorientable <Klein bottle>; the other rows are fixed by <Euler characteristic> and the <classification theorem for surfaces>.
Explicit square representatives are
$$
\boxed{S^2:aa^{-1}bb^{-1},\quad\mathbb{RP}^2:aabb^{-1},\quad T^2:aba^{-1}b^{-1},\quad K:aba^{-1}b.}
$$
All edge interior points have disk neighbourhoods by joining two half-disks, and each identified vertex has a cyclic sequence of corner sectors, again a disk neighbourhood. There are no leftover boundary edges. The enumeration therefore describes closed surfaces, and the twelve signed pairings leave no additional cases.
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2015-square-surface-pairings.png]
{title=Four closed surfaces obtained by pairing the sides of a square}
{description=Arrows and labels specify complete side-identifications. The examples realize the sphere, real projective plane, torus and Klein bottle; vertex counts and orientability distinguish their types.}
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