Mem is the first letter of the word Melech, king, and is the middle letter of the Hebrew alphabet. The open Mem represents the revealed aspects of the Torah, whil the closed Mem (the end-letter) represents the secret aspects of it. (Tractate Shabbat 104a, See Raskin, p. 129) It's central role is also reflected in Hebrew words for many of the fundamental parts of nature, such as Mother (Em), Water (Mayim), Shamayim (Sky), and Shemesh (Sun). The Mem also represent the womb. (Raskin, p. 130) Mem, along with Alef and Shin, is also known as one of the "mother-letters" in Kabbalah.
Nun is the first letter in the word, Nefilah, fall, and stands for the Divine attribute (sefirah) of Malchut, Kingship, and is related to one of the names for Mashiach: Yinon. (Likutei Moharan) Nun also means "fish" in Aramaic, and is associated closely with Moshe Rabbeinu. Joshua is called Yehoshua Bin Nun, the son of Nun, because a teacher (in this case, Moshe) is like a parent.
In Kabbalah, the sun is connected to Chochmah, wisdom. The moon is Malchut, as it reflects the light of the sun, in a way that can be absorbed by the naked eye. This phenomenon also very much represents the idea of the Oral Torah. Talmud teaches that Moshe was like the sun, while his disciple, Joshua, was like the moon. (Bava Batra, 75a)
The Sun is saying: "The sun [when covered by] the moon, stood in its abode; they speed at the light of Your arrows, and at the shining of your glittering spear." (Habakuk 3:11)
The Moon is saying: "He made the moon for the festivals; the sun knows the time of its coming." (Psalm 104:19)
It is fascinating how the song of the sun refers to the moon, while the song of the moon refers to the sun. The sun and the moon represent the ultimately duality, often found in many aspects of nature. The Talmud teaches that at first, both the sun and the moon were the same size, and it was exactly the aspect of kingship regarding them that led to a potential conflict: "The moon said to the Holy Blessed One, 'Sovereign of the Universe! Is it possible for two sovereigns to wear one crown?' God answered, 'Go, then, and make yourself smaller.' (Chullin 60b)
Shevat is very much a celebration of nature. On Tu B'Shvat, we celebrate the New Year of the Trees. As explained previously, Shevat is also very much associated with the Oral Torah.
In the calendar, the sun determines the days, weeks, and seasons, while the moon determines the months. It is around this time of year that the cycles of the sun and the moon have to completely align. That is why often, 7 times every 19 years, the Jewish calendar contains not one, but two months of Adar.
Adar is also very much about duality, particularly the holiday of Purim, which contain so many pairs, sometimes complementary and sometimes antithetical: Mordechai and Esther, Haman and Achashverosh, Mordechai and Haman... Purim, and the mitzvot associated with the day, are also very much tied to the letter Mem: Mishloach Manot, Matanot L'Evionim, Mikrah Megillah, etc. Finally, Adar and Purim in particular is associated with the Sha'ar HaNun, the 50th gate.
Adar is the month of Pisces (fish, Nun). The 7th of Adar is also the date of the birth, as well of the passing, of Moshe Rabbeinu.