For most of Christian history, women covered their heads for corporate worship in observance of the apostle Paul's instruction in his first letter to the church of Christ in Corinth.
Keeping in mind the exceptions, as written Christian history and, notably, Christian art evince, covering was practiced generally across the Christian spectrum into the modern age.
Indeed, the Catholic Church once required women to celebrate the Mass with veiled heads:
"Men, in a church or outside a church, while they are assisting at sacred rites, shall be bare-headed, unless the approved mores of the people or peculiar circumstances of things determine otherwise; women, however, shall have a covered head and be modestly dressed, especially when they approach the table of the Lord." (1917 Code of Canon Law. canon 1262)
In 1983, Pope John Paul II issued an ecclesiastic pronouncement that replaced the 1917 Canon Law, the updated 1983 Law including no such command regarding head-coverings.
Here I will repeat the argument of the Catholic Answers that (1.) further Catholic law states that any law not re-issued in the 1983 update is "abrogated" and (2.) since the 1970's the official Catholic doctrine has essentially been that "these ordinances, probably inspired by the customs of the period, concern scarcely more than disciplinary practices of minor importance, such as the obligation imposed upon women to wear a veil on their head (1 Cor. 11:2-16); such requirements no longer have a normative value" (see Inter Insignories).
Indeed, that it is no longer required for Catholic women to veil their head is not the controversial point (though women must veil in the older form of the Catholic Mass) and does not bear further evidencing.
So, while Catholic doctrine has changed, what of the Bible's teaching on the subject?
Paul straightforwardly commends the Christian congregation in Corinth in Ancient Greece for "remembering me [Paul] in everything and for holding to the traditions just as I passed them on to you" (1 Cor. 11:2).
Yet, he has counsel for them, which must be presented in full:
3. But I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God.
4. Every man who prays or prophesies with his head covered dishonors his head.
5. But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved.
6. For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head.
7. A man ought not to cover his head, since he is the image and glory of God; but woman is the glory of man.
8. For man did not come from woman, but woman from man;
9. neither was man created for woman, but woman for man.
10. It is for this reason that a woman ought to have authority over her own head, because of the angels.
11. Nevertheless, in the Lord woman is not independent of man, nor is man independent of woman.
12. For as woman came from man, so also man is born of woman. But everything comes from God.
13. Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered?
The problem is that of the teachings the Corinthians have held to, they have otherwise neglected that women should cover their heads when praying and prophesying.
As a remedy, Paul re-emphasizes the importance of following this teaching, giving two arguments about woman being the glory of man and a further argument "because of the angels."
The first point is relevant insofar as the head-covering serves as a fitting veil of the glory of man in woman and the glory of a woman's hair (v15), so God's glory may be all the more clear when a woman is praying or prophesying.
The second point about angels is open to endless speculation, but it must be noted these points are not "customs of the period", but extra-cultural concerns.
The issue is not whether a woman's hair is her covering, (previous Catholic teaching contradicts this idea), or a where a woman should cover (just in the congregational assemblies?), but is Paul's command here a normative rule still in force for all Christian women (and men, who should not cover) everywhere today?
The answer is yes because Paul roots Christian covering in factors independent of the passage of time or changing of cultural contexts.
Furthermore, while the Catholic Church may think Paul's command was "of minor importance", the fact he included it in his letter, taking the time to develop an argument in its favor, suggests it was of more than minor importance, despite its widespread neglect today.
I wonder if women had arrived for Mass in the 10th century with uncovered heads making this same argument if they would have been received well.
Most Protestants, also ignoring this command, may miss the casualness with which this Bible teaching has been cast aside by the Catholic Church, especially in light of its insistence that Paul's commands about women teaching and having authority over men in the church, and speaking in the assembly are "of a different nature", thus not cultural in nature, and in full force today (see the previously linked document).
Paul mandated that the Corinthians obey the Christian tradition on the subject, which was in place in all the churches (v16), and grounded in extra-cultural concerns.
While we should never "major in the minors", we should be careful about which Bible teachings we brand as "minor" and thus freely abrogate with the stroke of an ecclesiastic pen.
Fr. Raymond Burke, a high-ranking Catholic clergyman, in defending this change by the Catholic Church, reasons that because head-covering had lost its significance, that it could mean different things to different people (i.e., send mixed messages), and "because we find that in all areas of the Church's life not requiring a distinction of sex, men and women today participate equally in the Church as baptized persons", we can further affirm the prudency of the Catholic Church on this decision.
The problem with his logic is that the very same things could have been said in Paul's day.
Head-coverings were common in the first century Ancient Near East, so the "sign" of covering for a female Christian would have meant nothing to a non-Christian outsider unless it was explained to them.
Moreover, not covering was apparently an issue when Christians were coming together (thus the issue of praying and prophesying), so if the sign was losing significance, it was among Christians!
And Paul's response?
In addition, the "different things to different people" argument is what allows groups like the Salvation Army to forgo baptism and communion totally to avoid the problematic, modern connotations they see in these "signs" and to replace them with other things.
Also note that the early church was pegged as sexually debauched because of pagan misinterpretations of their "love feasts" and as cannibalistic because they ate the "body and blood" of their founder.
Yet, they continued on.
The reality is that being a Christian, with all that entails, will often run counter to culture and the proper response is never capitulation when an issue of inspired doctrine is in the balance.
Finally, men and women can be equal but different (as in regard to one wearing a covering and one not), a fact which Burke concedes when he mentions that Catholic women cannot serve in the priesthood of their church.
In the end, the Catholic teaching on head-coverings, when weighed, comes up wanting and we are reminded that while the doctrines of men may change, God's Word is settled forever in Heaven (Psalm 119:89).