Does your period affect your immune system?

I don’t really know where to start, other than the obvious: does your menstrual cycle affect your immune system?

I thought I was making something out of nothing, but I’ve realised every time I have my period, I’m really ill. A mild cold will take ages to get over, aches and pains and susceptible to infections quite quickly. I should note that I recently found out that I’m anaemic with heavy periods so that could explain why, coupled with my heavy flow and anaemia, my immune system is compromised.

I’m on iron supplements and I’m not a huge fan of them for several reasons, and I should be taking them more often, but the side effects for me aren’t worth it. Not only do I have to deal with a heavy flow, which is very regular, I have to deal with whatever cold I’ve caught, and constipation from the supplements. It feels like I get one good week a month where everything is ‘normal’ - outside of that, it’s PMS and colds.

I should note that I’ve been to the doctors about my observations and it was dismissed. They suggested allergies (like seasonal allergies, which I don’t think I have. I haven’t found the source of an allergies, just constantly sneezing like a normal person would) and scarily, birth control to stop the periods to see if my symptoms respond to that. I’m not on, and never have been on BC. I’m in my mid twenties.

Does anyone experience anything similar?