Is there any relationship between menstruation and either gassiness or diarrhea?

Yes: The hormonal changes associated with the menstrual cycle have been associted with changes in bowel habits and are a common cause of irritable bowel type symptoms.

Prostaglandins: In addition chemicals called prostaglandins are released with your period. Meds like Ibuprofen are used to block them. These prostaglandins may also cause diarrhea.