a man may reincarnate into the body of a woman

שער הגלגולים, הקדמה ט'
גם דע, כי לפעמים יתגלגל האיש בגוף נקבה, לסבת איזה עון ...

Sha'ar Hagilgulim, Chapter 9
Sometimes a man may reincarnate into the body of a woman because of a sin, (cultural context) …

R' Yosef Kaaro - Shilchon Aruch רבי יוסף קארו - המחבר 17th Century Ottoman Palestine

מגיד מישרים ח׳:ג׳

(ג) הלא אודעתך בשבת שעבר רזין דתרין נשי קמיית', והשתא אתינא לאודעותך רזא דהאי איתתא תליתאה, הלא לך למינדע דהאי אתתא הות בזמן עבר זכר ת"ח כשר אלא דהוה כילי בממוניה ולא הוה עביד מיניה צדקה, וגם הוי כילי בחכמתו ולא היה רוצה ללמד לאחרים ועל כן נענש להתגלגל בנקבה מדה כנגד מדה הוא לא היה רוצה להשפיע לאחרים כעין אותם שמזכירים אותם לגנאי כי על הראשונים נאמר זכר צדיק לברכה כלומר הצדיק הוא משפיע לאחרים הוא גורם שצדיק יסוד עולם ישפיע בכנסת ישראל, וזה שאמר זכר כלומר שיסוד הוא גורם שיעשה פעולת זכר, ולפיכך אינה יולדת בחברתך אבל אני אעשה שיתנוצצו בה נצוצות מנשמת נקבה, וזה סוד ויתן יי לה הריון ותלד בן האמור ברות מלמד שלא היה לה עיקר מטרין הוא הדבר שאמרתי שנשמתה היה נשמת זכר כי היא היתה תמר, וכבר אמרתי שנשמת תמר היתה נשמת זכר. והנה סיבת עכבת לידת אשתך זהו בסבת שנשמת שניכם נשמת זכר ועכשיו כבר התנוצצו בה ניצוצות נשמת נקבה ועל ידי כך תקבל הריון ממך, וזכתה לכך ע"י מעשיה הטובים ועל ידי הצער שנצטערה על ידי שגלית מאצלה. וגם ע"י הצער שנצטערה בשירות חולייך כי כל כך זוכה כשיסורים באים עליו כשסובלם בסבר פנים יפות כמו שזוכה בעשיית מצוה. והטעם לפי שעל ידי היסורים בגוף מתחלש כח הטומאה ומזדככת הנפש מטומאת זוהמת הנחש ונשארה טהורה ונקיה, והנה נתנוצצו בה ניצוצות הללו מנשמת בת זוגך, ועד עכשיו לא היה אפשר מפני שהיתה נשואה לאחר ועכשיו נתאלמנה, וע"י הזכיות הנז' נתנוצצו ממנה ניצוצות בזוגתך זאת כדי שתוליד ממנה כאשר אמרתי וע"י הדברים האלו הנזכרים זכתה להתנוצץ בה ניצוצות נשמת נקבה ועל ידי כן תלד לו בנים זכרים כאשר הבטחתיך ואמרתי לך ואתה שלום:

Maggid Meisharim 8:3

I have already revealed to you last Shabbat concerning your first two wives. Now I have come to reveal to you the secret your third wife.
You should know that this woman was in the past, a proper male Torah scholar. However, he was stingy with his money and would not give charity. He was also stingy with his wisdom and would not teach others. He was therefore punished that his soul migrated into a woman, measure for measure…
Therefore, his soul was incarnated into a female, who is constantly receiving and needs someone to bequeath to her. Therefore you see, that she does abundant charity and loves you very much because you work to spread Torah and toil in writing books to teach others…because these things bring about the rectification of her soul, she therefore loves you…
It is because she has the soul of a male that you have not had children from her, because a male and another male cannot produce offspring. If you shall point out that she has children from her first husband, this is because the first husband has the spark of a female soul within him…
(The magid explains at length above, that it is entirely possible for a male to have a female soul and vice versa.)

Baal HaTanya

Torah Ohr, Volume 1, Parshas Vayigash, page 86 in the standard Kehot edition

Talks about how the higher source becomes lower when put in this world. Also the higher depends on the lower. Thus the grass depends on dirt but not the reverse because the grass is higher. You see in this some reflection of the higher source of the dirt even though it is lower in this world.

R' Hirsch and Magen Avraham

Rav Hirsch says women have more enthusiasm for their mitzvos. This doesn't mean they would have more for the men's mitzvos. Indeed we see in the Zies Ra'anan (Magen Avraham) that women are exempt from positive time bound commandments because their yetzer tov is smaller and if commanded would likely not be motivated to perform them.

source

What has a higher source is lower in this world. (Likutei Torah 2:34c) And so the Maharal, upon whom much of Chabad is based, says repeatedly that women are on a lower level in this world. See Maharal: Chidushei Agados, Makos 23b; Chidushei Agados, Nidah 45b; Derech Chaim 1:5; Derech Chaim 2:9;Gur Aryeh, beginning of parshas tazriah; Tiferes Yisrael 28; Tiferes Yisrael 4.  For example, Tiferes Yisrael 28: "The woman is more physical and her level is not as high as is the man’s. Therefore Adam HaRishon was commanded first since he was closer to Hashem. The woman’s greater distance from Hashem led to her initiation of the first sin and destruction upon the world. " He says also men excel in Chochmah and women in Binah. But the Tanya taught that Chochmah is higher than Binah. (Today's Tanya Lesson, Shevat 10, 5779 · January 16, 2019, Likutei Amarim, Chapter 19)

Commentators

Commentator	Source	Summary
Abarbanel	Bereshis, Chapter 1
Akeidas Yitzchak	Chapter 6	Woman is not commanded in the positive time-bound commandments because she is weaker than the man.
Baal Shevet Musar	Midrash Talpiyos, Ohs Aleph, Anaf Isha	The man is above the woman since his soul is from fire and air. Her soul is from the more earthly elements of water and earth.
Bartenura	Mishnah Horarios 3:7	A man's obligation in more mitzvos shows that he is more sanctified than the woman.
Chasom Sofer	Gitin 57b	Women are quicker to do teshuvah.
Lubavitcher Rebbe	Sichos in English, Iyar-Tammuz 5744, Vol. 21, pp. 69-72	Women are exempt in order to take care of the home. When the man performs the positive time bound mitzvhas his wife gets included in his actions since they are as one. Before marriage, this is taken care of by the destined partner.
Magen Avraham	Ziet Ra'anan on Yalkut Shemoni, Shmuel 1:1	Woman is exempt from positive time-bound commandments because her yetzer tov is smaller. If she were commanded, it is likely she would not comply.
Maharal	Chidushei Agados, Makos 23b	The man is created more in Hashem's image than the woman.
Maharal	Chidushei Agados, Nidah 45b	Women have bina yesayrah but men have chochmah yesaryah. The former is more ethereal and designed for taking care of the home. One should not rely on it for spiritual matters as the man's intelligence is better geared for that. "The soul-faculties of “knowledge” (Daat) and “understanding” (Binah) are lower than Chochmah; yet the level of Chochmah is prevented from acting upon them (and upon the other, still lower, faculties) as long as they are immersed in mundane pleasures. Thus, the Chochmah of their divine soul is dormant, not dead. It has lost none of its potency, only its ability to exercise it; just as when one sleeps he retains full possession of his faculties though he cannot use them." Today's Tanya Lesson, Shevat 10, 5779 · January 16, 2019, Likutei Amarim, Chapter 19
Maharal	Derech Chaim 1:5	The woman is more physical than the man. He is considered the spiritual form relative to her.
Maharal	Derech Chaim 2:9	The man has more 'powers of the soul, while the woman leans more towards the physical. The woman has a leaning towards witchcraft because of her physical, lower level.
Maharal	Gur Aryeh, beginning of parshas tazriah	Adam was created after Chava indicating his higher spiritual level.
Maharal	Tiferes Yisrael 28	Chava was more physical than Adam, on a lower plane, and less connected to Hashem. Adam was given the first commandment because Chava would not have been able to work with only a single commandment due to her lower level. Women were spoken to first at Har Sinai because with a full set of commandments there was less chance of them stumbling as Chava did and speaking to them first served as encouragement for bringing their husbands and sons to the beit midrash.
Maharal	Tiferes Yisrael 4	The man is the spiritual form and the woman the matter. Positive commandments involve positive spiritual action. Therefore, they are more relevant to the man. She is equal in the negatives because she can do as much harm and the negatives prevent against harm.
R. Moshe Feinstein	Igeres Moshe, Orach Chaim IV, 49	Men and women are equal in holiness.
R' Samson R. Hirsch	Vayikra 23	Women are obligated in nearly as many positive commandments as they are exempt. Nevertheless, they are exempt because they have more fervour and more faithful enthusiasm for their calling and are in less danger to temptation since they are shielded from the world.
Rambam	Mishnah Horarios 3:7	The man is more sanctified than the woman.
Rav Tzadock Rabinowitz	Dover Tzedeck p. 119	Adam was created with Hashem's decree alone but Chava was created with Hashem's decree and Adam's desire. Therefore, Adam is on a higher plane than Chava.
Taz	Shulchan Aruch, Orach Chaim 46	The morning blessings "Has not made me a ..." work in sequence of lower to higher levels.
Vilna Gaon	Even Shelaimah 1:8	The woman struggles more to do good since she is created closer to the Satan and to sin.

Response to Chabad person who was slinging the nonsense

What has a higher source is lower in this world. (Likutei Torah 2:34c) And so the Maharal, upon whom much of Chabad is based, says repeatedly that women are on a lower level in this world. See Maharal: Chidushei Agados, Makos 23b; Chidushei Agados, Nidah 45b; Derech Chaim 1:5; Derech Chaim 2:9;Gur Aryeh, beginning of parshas tazriah; Tiferes Yisrael 28; Tiferes Yisrael 4.  For example, Tiferes Yisrael 28: "The woman is more physical and her level is not as high as is the man’s. Therefore Adam HaRishon was commanded first since he was closer to Hashem. The woman’s greater distance from Hashem led to her initiation of the first sin and destruction upon the world. " He says also men excel in Chochmah and women in Binah. But the Tanya taught that Chochmah is higher than Binah. (Today's Tanya Lesson, Shevat 10, 5779 · January 16, 2019, Likutei Amarim, Chapter 19)

Higher source

Most of the material I put out here is straight up Torah sources. I don't give my opinion too much. But I am going to do that now to try to explain what seems a contradiction. Why do the middle acharonim talk about men being on a higher level but contemporary gadolim talk about an equality? How can both be true. On top of that the Kabbalah says women come from a higher sefira, that of malchus.

So I'll be brief here because I'm typed out, I think maybe it's like this. What's higher in the pre-world is lower here. When a stone wall falls, the stones up top roll the farthest. Personally, I feel there's something spiritual about women and something unspiritual. So on the spiritual side you have the emunah peshuta, the compassion (much less in our era though), the morality in many areas, the refinement of personality, the absence of violence. On the not spiritual side you have gossip, materialism, and irrationality.

So I think one way of thinking of it is that the Maharal and other Acharonim are explaining this world where the higher became lower, so the male is on a higher plane. However, because of the higher source of the female, the spirituality will come out particularly in the material plane where the spiritual is also manifest as Chasidus notes. So physically in many ways the woman is more spiritual. She can make babies. She is more physically refined, elegant, where the male is something of an ape. I think this explains feminine beauty. There's something spiritual there. And this is one reason it needs to be covered! And this is why her emunah is an emunah peshuta. It's sort of amorphous, basic, hard to define. But with the higher human functions the male is more spiritual. He is more rational, which is a spirituality. He sins less with his mouth, speech being a higher function. This is why he has more mitzvos. His spirituality is more revealed so it can be expressed in mitzvos. Her spirituality is less defined, more general, basic so she lives on a more intuitive plane which doesn't translate as well to positive time-bound mitzvos. (I have noted before that positive time bound mitzvos are masculine - sitting outside in the succah, wearing the same leather straps every day, etc.)

So as each is spiritual in different ways, you can say both are equal. So the way the scholars speak depends on how you look at it.  You can say he is higher. You can say she is sort of higher. You can say both are equal overall.

In this world, he needs to be in charge but with great respect for her intuitive spirituality that often reveals a higher spirituality and without which he could not function.

After Moshiach comes, when the sin of Chava is rectified, it could be that everything switches, and the female emerges. The Baal HaTanya talks about this. Feminism could be a sort of evil manifestation of that, just like all the technology is an evil manifestation of the Torah that was supposed to come into the world in modern times. R' Kessin talks about this on his lashon hara lectures, that lashon hara caused the divine spirituality to be redirected to the physical.