You should have a look at the paper ([now broken Springer link](http://www.springerlink.com/content/xkf2bqn5p7bndcrd/))
which deals with nonorientable surfaces with boundary.

**Edit** (April 17, 2023) I don't remember what paper I had in mind at that time. Sorry for such a poor informational link; Springer sucks. Most probably it was the following one: Korkmaz, M. Mapping Class Groups of Nonorientable Surfaces. Geometriae Dedicata 89, 107–131 (2002) [Link at Springer site](https://link.springer.com/article/10.1023/A:1014289127999), [DOI link](https://doi.org/10.1023/A:1014289127999).