= Solution
An <orientable surface> admits local coordinate orientations that are consistent on overlaps. Equivalently a <triangulation> can be oriented so that adjacent faces induce opposite directions on their common edge. Use a polygon word to record its successive boundary edges, identifying equal letters in their indicated directions; a negative exponent reverses that edge direction.
Square models are \b[$P:abab$, $K:aba^{-1}b$, and $T:aba^{-1}b^{-1}$]. The first identifies antipodal boundary points of a disk, giving the <real projective plane>. Only the <torus> among these three is <orientable>: its paired sides traverse the boundary oppositely, whereas each other model has a same-direction pair. The projective model has two vertex classes, and the other two one, giving Euler characteristics one and zero respectively.
The other possibility is \b[the <sphere>], for example the word $aa^{-1}bb^{-1}$ with three vertex classes. There are no others: a paired square has one face, two edges and between one and three vertex classes, so its <Euler characteristic> is $V-1\in\{0,1,2\}$. The <classification of compact surfaces> leaves precisely the <sphere>, <real projective plane>, <torus> and <Klein bottle> at those values, with the indicated orientabilities.
For connected sums, remove a disk from each <surface>, join the resulting boundary circles, and cut a joining arc to produce a single polygon. Its word concatenates the two original words. The <real projective plane> also has the bigon word $cc$, so
$$
\boxed{T\mathbin{\#}P:\ aba^{-1}b^{-1}cc,\qquad T\mathbin{\#}K:\ aba^{-1}b^{-1}cdc^{-1}d.}
$$
These use respectively six and eight sides, proving the requested hexagon and octagon constructions. Their Euler characteristics are $-1$ and $-2$, also obtained from $\chi(X\#Y)=\chi(X)+\chi(Y)-2$.
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