Some people are offended by the sensibilities of two scriptures by Paul which seem to note a secondary status of women in relation to men. This is my attempt to analyze one these two scriptures.
First of all, for context, Paul is here either exhorting women to cover their heads during religious practices or men not to cover their head during religious practices, or both. Some studies have focused on the less traditional idea that the main focus ought to be on the man not covering his head while praying, which was against Jewish tradition. Most however focus on the concept that a woman ought to cover her head, even though the church services at that time were in the home, which was a territory were it was culturally acceptable at the time for a woman to have her hair down. Paulâ€™s reasoning here seems to be along the lines is that when a home is used for worship it should be treated as a public forum, where at the time it was considered flirtatious for a woman to flaunt her hair (and paradoxically, a sign of immorality to have short hair as a woman.)
The concept here specifically in question is the usage of head in vs. 3:
11:3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. 
So what is meant by â€œheadâ€ here. It is kephale in Greek, which denotes â€œthe head (as the part most readily taken hold of), literally or figuratively:–headâ€  Many claim it denotes a subordinate status for women in either the culture the bible was written in, or in the culture of God. They take head to mean â€œsuperior.â€ There is an alternative understanding however. One Author notes:
In the Greek language of NT times, “head” (kephale) did not necessarily serve as a metaphor for someone in a position of authority (as it normally does today). In ancient Greek thought, the heart, not the head, was considered to be the seat of a person’s reasoning and decision-making powers; the head was generally regarded as the source of life for the body. Such an understanding of the head metaphor is clear from the description of Christ as the source of life, health, and strength for his body, the church, in Colossians 2:19 and Ephesians 4:15 (just prior to where the husband is called the head of the wife) … The metaphorical meaning for “head” in this passage seems to be that of “source” or “origin”; this meaning still exists in modern English (we sometimes speak of the head of a river), though it was more common in ancient Greek. Christ is referred to as the head of the church in this sense in Colossians 1:18. 
This concept of the source of the woman being the man seems to garner support from directly from vs. 8 and 12 where â€œThe woman is of the man.â€, and is apparently meant to call upon the idea that Eve came from Adamâ€™s Rib. This idea is not without itâ€™s skeptics:
Any honest appraisal of 1 Corinthians 11:2-16 will require both teacher and students to confront the patriarchal implications of verses 3 and 7-9. Such implications cannot be explained away by some technical move, such as translating kephale as “source,” rather than “head,” [meaning superiority] because the patriarchal assumptions are imbedded in the structure of Paul’s argument…
So what Patriarchal Assumptions are imbedded in verses 7-9? The same author elsewhere notes:
In verses 7-9, however, Paul raises the theological stakes by introducing a new line of argument based on his reading of the Genesis creation story. A man should not cover his head because man is created as “the image and glory of God” (Gen 1:27), but woman is “the glory of man.” Here, regrettably, Paul gets himself into a theological quagmire. Genesis 1:27 explicitly says that humankind is created “in the image of God … male and female he created them.” Paul’s interpretation of the text, however, seems to depend on a tradition — perhaps based on Genesis 2:7 — that thinks of the male only as originally created in God’s image…
My reading is somewhat different here.
7 For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
This is easily the most problematic of the three verses. That being the case, I will develop thoughts on it more fully in a moment.
8 For the man is not of the woman; but the woman of the man.
This verse supports the concept of â€œheadâ€ as source without any deeper analysis.
9 Neither was the man created for the woman; but the woman for the man.
The word here translated as â€œforâ€ is dia, which Strongâ€™s defines thus: â€œa primary preposition denoting the channel of an act; through (in very wide applications, local, causal, or occasional)â€ Thus woman is only â€œforâ€ the man in the causal sense, and not in some sense of subordinate ownership. If it wanted to imply woman was for the sake of man, it would probably have used huper, as it does in vs. 24-25 of the same chapter. In any case, this verse only further supports the idea of the head as the source, and not necessarily as the superior being.
Returning to the problems associated with verse 7, which in the NRSV is translated as â€œFor a man ought not to have his head veiled, since he is the image and reflection of God; but woman is the reflection of man.â€, it may do well to associated this verse not with verses 8 and 9, as being associated with Adam and Eve and their biblical conception, but rather with verses 4-6, which emphasize instead the cultural believes of the local people at the time. 4-6 say that while typically a Jewish man would cover his head in prayer, Paul explains it is culturally inappropriate in Corinth. Further a Woman who prays without her head covered could be seen as immoral cultural based on the social norms. Perhaps Verse 7 is saying that in the cultural lense of the people of Corinth, the man is the reflection of Deity, and the Woman the reflection of the man. One reason to support this view is the chiastic structure of the text, which is as follows:
A__Introduction(2-3)- Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you. But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
B___’woman,’ ‘uncovered,’ ‘to pray,’ ‘man,’ ‘glory’ (4-7)- Every man praying or prophesying, having his head covered, dishonoureth his head. But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered. For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
C____(8a) - For the man is not of the woman;
D______(8b) - but the woman of the man.
E________(9a) - Neither was the man created for the woman
F__________(9b) - but the woman for the man
x____________(10) For this cause ought the woman to have power on her head because of the angels.
F’_________(11a) - Nevertheless neither is the man without the woman,
E’_______(11b) -neither the woman without the man, in the Lord.
D’_____(12a) - For as the woman is of the man
C’___(12b) - even so is the man also by the woman; but all things of god.
B’__(13-15) ‘woman,’ ‘uncovered,’ ‘to pray,’ ‘man,’ ‘glory’ - Judge in yourselves: is it comely that a woman pray unto God uncovered? Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
A’_(16) Conclusion – But if any man seem to be contentious, we have no such custom, neither the churches of God.
If this Chiasmus holds true, then the real question of whether or not a woman is subordinate to man, under God, does not rest in verse 7 at all, but in verse 10. Due to the variety of opinion among scholars as to the meaning of verse 10, it is difficult to put forth only one opinion as to the meaning of the text.
Strongâ€™s gives the Greek as follows
dia touto ofeilei h gunh exousian ecein epi thV kefalhV dia touV aggelouV
Through [dia] this [touto] owes/should [opheile] the [he] woman [gune] authority/power/liberty [exousian] to have [echein] on [epi] the [tes] head [kephales] because [dia] of the [tous] angels [aggelous].
Many different bibles have taken this data and translated it differently.
The NAB translates it as â€œa woman should have a sign of authority on her headâ€. Some view this sign of authority as the sign of manâ€™s authority over the woman, but I do not believe this contextually makes any sense, as this sign allows the woman to pray and prophesy, exactly as the man is doing. I believe the veil here discussed is used as a symbol to remove any concept of superiority or subordination, as the sign of authority would demonstrate the equal status of the woman with the man in praying and speaking under inspiration.
The newly popular ESV follows the same reasoning, but contains an alternative reading of the term angels which is interesting to note. It notes that angels here may actually mean â€œmessengers, that is, people sent to observe and reportâ€. In other words, it is important for the members of the church to act within the social norms because people from the typical people of Corinth may be checking in on them.
The NRSV also follows this same reasoning, but also offers an alternative reading that the woman ought to have â€œfreedom of choice regarding her headâ€. This goes back to the concept that the real emphasis in this scripture is that men should not wear head coverings in Church, while women are free to choose for themselves whether the should wear a head covering in church. For some reason, this concept reminds me of growing up Catholic and seeing all the women in their Easter hats or bonnets. This reading obviously does not put woman in a subordinate status to man.
To conclude I will paraphrase 10 and link it to verse 11 and 12. Women should wear the customary symbol (but are free not to), in order to denote they are equal in authority, independent of man. Nevertheless, neither is the husband without the wife, nor the wife without the husband, in the Lord. Also, As Eve came from Adam, now all men are born from women, but all are born of God.
And I would add, All were born and are reborn equal and free, through the atonement of Christ.
 for more information, see footnote 331
 unless otherwise noted, scriptures reference KJV throughout
 Gender Equality and the Bible
 A reference guide to 1 Cor. 11
 This concept of the Chiasm loosely follows the one here.
 ESV available online here.