Apparently, there's this belief out there, parially encouraged by such well-respected theologians as Dan Brown (author of 'The Da Vinci Code') and Leonard Nimoy (patron of the 'Shekhina' pornography, er, art exhibit[ionism]) that the term Shekhina is some sort of Godess-type figure, the female counterpart of God, who was part of ancient Israelite worship, only to be repressed until She was 'rescued' by contemporary new-age ecofeminist Judeo-Wiccans. Yet another example of how the truth is stranger than fiction.
Since this and next week's Torah portions specifically deal with the Mishkan, and express the telos of the Mishkan as being 've-shakhanti betokham' and 'le-shokni be-tocham', I felt it appropriate to spend some time formulating what I think 'Shekhina' is, and why it's associated with femininity.
Man can't understand why God would create this world. We can't know what He would possibly get out of it. All we can know is that He did create it, that He has goals for it, and that human free will is the mechanism by which those goals can be realized. Human free will is the element which makes this world different. The ability to choose between good and evil, to make to world more or less Godly, lies with man alone.
Chazal expressed God's impetus for creating the world as a 'ta'avah', a lust, an urge. "Nit-aveh lo l-HKB"H lihyot lo dira ba-tachtonim" - "God had an urge' to have a home in the nether worlds" (not 'an apartment in the underwear', which is what it would mean in modern Hebrew).
It's a restatement of a verse "Ivah le-moshav to" - "He 'lusted' a dwelling of his own". It's also expressed in another context in Chazal, who state that God 'has an lust' for the prayers of the righteous. Lusts and urges are not rational desires. The term 'Tayva' has a strong connotation of the sexual impulse - something irrational but intense. To speak of God this way seems blasphemous, yet our Sages do speak this way. I believe that they are trying to emphasize the non- or pre- rationality that characterizes God's motives. Reason is one of God's creations. It does not determine Him, He determines it. Thus, for us humans, God's impulse to create this world is, for all intents and purposes, an 'urge'.
However, this 'urge' puts God at a disadvantage in this world. If his desire is for humans to autonomously choose Godliness, then He is limited in His ability to intervene in man's decisions. He needs man. God relies on man for His expression in this world - it is man that must build God's 'dwelling space in the nether worlds'. Otherwise, the human realm is no different from the 'animal' or 'angelic' world - of either pure instinct or pure intellect, but no automonous choice. It's an interesting reversal - God waiting for us, God 'exiled' from the world that he created, waiting for man to redeem Him. Every person constructs a perceptual world of his or her own, and assigns God a greater or lesser role in that world. The same is true of each community, each culture, and each nation; God is assigned a particular but ever-changing role within the collective or shared perceptions of the group. Within individual perceptions, there goal is to construct a reality in which everything is viewed as a manifestation of Godliness, or God's will. The Torah is God's revealed will, which man, by use of his intellect, applies to this world. The Mitzvot, the Halakha, is the way in which we actively apply Torah to our reality. Worship, Prayer, and Brachot are the tools by which we train ourselves to perceive God within our world. It breaks down more specifically, but acamo"l.
Nevertheless, this is not the sum of God's involvement in our world. There are occasions, which have diminished over time, in which God actively tinkers with history, or continues to create and sustain the world. However, like a parent, God gradually takes a less and less prominent role in the lives of His children. Thus, the relationship between God and man remains a complicated one. Sometimes God is the active partner, and sometimes man. OVer time, the balance of the relationship has shifted toward man's greater activity. Richard Elliot Freidman has a great exposition of this phenomenon (which was already noticed by Chazal) in his book The Hidden Face of God. Ayen sham, ve-acamo"l.
In the symbolism of Chaza"l and to a great degree, even in TaNach, the more active partner in a relationship is viewed as the 'masculine' partner, and the 'acted upon' (not 'passive') partner associated with femininity. As I've demonstrated, when it comes to God's actual 'presence' or 'dwelling' within this world - His 'Shekhina' (literally 'indwelling'), he is waiting for man to make his move. This aspect of God's relationship wiith the world is associated with femininity. Thus, the Shakhina is the feminine aspect of God's relationship with man. (We can't talk of a duality within God Himself, but His relationship with man is definitely complex and multifaceted). Man's failures result in 'Galut Ha-Shekhina' - God's alienation from His own world, in which He has no choice but to wait for us.
This and next week's Torah portion form the end of the story that began with creation - God's plan to dwell in this world (see Ibn Ezra and Ramban on 'le-shokhni betocham', in next week's portion). It gives man the recipe for building a structure around which a community, a nation, a civilization, a culture, in which God's presence is percieved and his will acted out - can be constructed. I believe this is the message of the 'palpability' of God's Presence in the Mishkan, which diminished with time, as man grew up and had to work harder to find God.