 Solution

A polygon with a complete set of side-identifications consists of a closed polygonal disk, a pairing of all its boundary sides, and a specified endpoint-matching homeomorphism for each paired pair. Form the quotient by these side maps, leaving interior points distinct. A convenient encoding labels paired sides by the same letter and gives them arrows; traversal in the arrow direction contributes $a$, and traversal against it contributes $a^{-1}$ to the boundary word. Every letter occurs exactly twice.

The orientations determine whether matched endpoints are identified in the same or reverse boundary-traversal order. This endpoint data is essential: pairing the same geometric sides with different directions can produce nonhomeomorphic surfaces. Subdividing the disk and the paired sides gives a triangulation of the resulting surface.