(I really wish Bible translators had a better term than "unclean". The concept has nothing to do with being dirty. It's more about association with death. I also wish theologians would quit equating atonement with salvation.)

"Blood of purifying" in #leviticus 12:4 uses the Hebrew word for ritually clean (tahor), so the blood removes or covers something that is ritually unclean (tamei). Does the blood of circumcision partly atone for the uncleanness of birth? A woman remains in tamei longer if she has a girl (80 days) than if she has a boy (40 days). Perhaps this isn't because girls are more "unclean" or anything like that, but because the circumcision of the boy allows that period of separation to be shortened. Speculation.

There are two or three blood atonements in Leviticus 12 after the birth of a child, depending on whether it is male or female:
1) Mother's blood
2) Circumcision (male only)
3) Sacrifice