Narcissus tazetta bulb extract has been marketed as an anti-ager and more recently (about a year ago) as a weapon against unwanted hair. An ingredients manufacturer, IBR, has made narcissus its schtick. It claims that the bulb must be harvested when dormant because that's when they are "restoring their youth, energy and beauty". That part sounds like codswallop to me, but the real point is that the bulb extract is supposed to stop cell proliferation and the same inhibitors will slow down the growth of unwanted hair.
Now the odd thing is, until I spotted Bliss Get Out of Hair, I haven't come across narcissus tazetta in a hair removal product. I did find it in a shampoo by Aveda, which to me seems like an oxymoron unless it's targeted at a rare demographic that wants to prevent hair growth on the head.
In addition to narcissus, Bliss' special trade marked formula Fuzzkill Complex also uses boehavia diffusa (a plant extract that is particularly good for treating guinea worm should one lay its eggs under your skin) and willow bark. The latter may indeed stop the itching because it is well-known inflammatory.