I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
[![Gluings of a hexagon][1]][1]
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?

  [1]: https://i.sstatic.net/LTbvQ.png