When I did my first TTC, I was told not to practice inversions during the menstruation cycle. By inversion I mean every pose where the heart is higher than the brain, so this would not be narrowed only to Sirsasana (Headstand) or Sarvangasana (Shoulder stand), but will include also postures as Uttanasana (Standing Forward Bend), Adho Mukha Svanasana (Downward-Facing Dog) or in Prasarita Padottanasana (Wide-Legged Forward Bend).
But, I have always done inversions in my personal yoga practice during every step of my cycle and have always taught them as well. I haven't feel in my personal practice the need to stop inversions postures. When I teach yoga, I tell women it is up to them to practice inversions, and I don't hold them long in class. I found, in my body and for my practice, they may have actually helped me reduce stress related symptoms.
Philosophically speaking, menstruation is considered to be apana, meaning that energetically, its vitality is downward-flowing. The argument against inversions during menstruation maintains that inversions will disturb this natural energetic flow. However, inversions are recommended in some systems of yoga as therapy to improve elimination of excess of apana. In Yoga: The Path to Holistic Health, B.K.S. Iyengar recommends practicing inversions to alleviate menstrual problems such as heavy flow and irregular periods.
One thing that resonated with me is that when menstruating the overall energetic flow of the body is downward and grounding energy, moving towards the earth. When we do a high energy yoga sequence, that may include inversions, the overall energy is upward. This made sense to me as my body on the first & second days of menstruation feel lethargic and like I am anchored to the earth, heavy as lead. Ultimately during this time the body is doing monthly processes and the energetic quality of these are downward and grounding. In the ayurvedic teachings even female bodyworkers are encouraged not to work and simply take rest on the heavy days.
Since I know of no studies or research that makes a compelling argument to avoid inversions during menstruation, and since menstruation affects each woman differently and can vary from cycle to cycle, I am of the opinion that each woman is responsible for making her own decision. So, I think it is definitely up to each individual woman to choose to practice inversions or not during this time, and especially during the first 2 days.