I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space: The first two give the torus and the Klein bottle, respectively. What are the last two? In particular, do their fundamental groups have another name? Do they have the same fundamental group?

EDIT: HJRW points out that I missed the several ways to glue a hexagon to get a non-orientable surface of Euler characteristic -1.