Inversions are very powerful energetically, and can reverse the downward flowing prana. This is why they should be done very lightly during menstruation, because the strong earthward flow of energy is very important and shouldn't be reversed. It is not about fatigue, lack of strength or any other issues. It is not gender-bias. Doing inversions like you normally would can prematurely stop the cycle...so unless you are consciously choosing to play with that as part of your sadhana, they should be minimized or avoided during the peak days.
During pregnancy, especially during the last trimester, inversions should be avoided, even mild ones like downward facing dog, to avoid moving the baby out of position or disrupting the powerful downward flowing prana (called apana). Inversions can be used as directed by a midwife to help the baby to move into position during the last weeks of pregnancy.