In this question : Two Fundamental Polygons for the Double Torus?
Lee Mosher says
There are four octagon gluing patterns (up to rotation and relabelling) which give a double torus.
It is a very interesting result to me, which makes me think about the following question.
How may patterns (up to rotation and relabelling) to glue a $4n$-gon to a genus $n$ surface?
I tried to find all above patterns for a $12$-gon and I failed, since there are too many possibilities.
I feel like my problem is a natural one, so I'd love to know if it's already been solved.