Bodies have genders (physical anatomy and genetics) and the brain has a gender (gender identity), but do the spirits/souls or our heavenly bodies have genders as well?
- Do we retain our gender in Heaven? at Christianity
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The Zohar writes that the Patriarch Yitzchok had the soul of a female.
Additionally, the Arizal explained that Michal wore tefillin because she had a male soul.
So it seems that souls do have genders, both male and female.
No. A soul is a spiritual entity and has no gender. However according to Kabbalah there are two worlds where souls can come from. A "Masculine" world and a "Feminine" world. The Lubavitcher Rebbe Z"L explains that that
the way of a soul from the "masculine side" is to transform the world by being pro-active and outgoing, while the path of the soul from the "feminine side" is to accept the world as it is, trying to live properly according to Torah, but without trying to influence the world.
So when the Kabbala uses those terms it does so as a metaphor to explain a concept not to indicate gender.
Note when explaining Michal the daughter of Shaul the holy Ari says that she had a
נשמה מעלמא דדכורא
loosely translated as "She had a soul from the world of the masculine (side)" Not that her soul was of the male gender.
I encourage you to read the whole article from the Chabad website - Female and Male Souls
the shemonei shearim shaar gilgulim (forget exact place) says that if a male soul is reincarnated in a female body, as sometimes happens, then that female cannot have a male child - only female children, as the verse says in parsha tazria "when a woman conceives and bears a male child", and that woman is really a man in a female body.
I think he says there that male soul are from adam while female souls are from chava.