You should have a look at the paper (now broken Springer link) 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, DOI link.
