Cardinal Raymond Burke, Prefect of the Apostolic Signatura, responded to an inquiry on this subject. While not a formal judgment of the Signatura, it reflects the opinion of the Church's highest canonical official after the Pope. Note in his answer that there is neither a canonical or moral obligation for women to use a head-covering. Even in the case of the Extraordinary Form there is merely "an expectation," whose failure to fulfil does not entail sin. image of original letter
4 April 2011
Thank you for your letter postmarked January 5, 2009, regarding the custom of the chapel veil. I offer you my sincere apologies for failing to respond to your letter, in a timely manner. I had placed your letter with some other papers. and have only recently discovered that I never responded to it.
The wearing of a chapel veil for women is not required when women assist at the Holy Mass according to the Ordinary Form of the Roman Rite. It is, however, the expectation that women who assist at the Mass according to the Extraordinary Form cover their heads, as was the practice at the time that the 1962 Missale Romanum was in force. It is not, however, a sin to participate in the Holy Mass according to the Extraordinary Form without a veil.
I wish you an abundant share in the strong graces of the Lenten Season.
Thank you for the assurance of your prayers for me. As a Cardinal of the Holy Roman Church, I have need of your prayers, now more than ever.
Invoking God's blessing upon you, while confiding your intentions to the intercession of Our Lady of Guadalupe, I remain
Yours devotedly in Christ,
Raymond Leo Cardinal Burke
Archbishop Emeritus of Saint Louis
Prefect, Supreme Tribunal of the Apostolic Signatura
Original FAQ from 2004
Canon Law The 1917 Code of Canon Law. canon 1262, stated,
1. It is desirable that, consistent with ancient discipline, women be separated from men in church.
2. 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.
When the 1983 Code of Canon Law was promulgated this canon was not re-issued; indeed, canon 6, 1, abrogated it, along with every other canon of the 1917 Code not intentionally incorporated into the new legislation.
1. When this Code goes into effect, the following are abrogated:
(1) the Code of Canon Law promulgated in 1917;
(2) other universal or particular laws contrary to the prescriptions of this Code, unless particular laws are otherwise expressly provided for;
(3) any universal or particular penal laws whatsoever issued by the Apostolic See, unless they are contained in this Code;
(4) other universal disciplinary laws dealing with a matter which is regulated ex integro by this Code.
Thus, there is no longer any canonical obligation for women to wear a head-covering, much less the more specific veil. Moral Law Given St. Paul's instructions in 1 Cor. 11:3-16 is there a moral obligation for women to wear head-covering, despite the revision of canon law?
Certainly, the moral obligation to dress modestly according to circumstances (e.g. approaching Holy Communion) has not been set aside. Modesty, however, can vary from place to place and time to time. As St. Thomas Aquinas explains, modesty concerns four areas of human behavior,
First, "the movement of the mind towards some excellence, and this is moderated by "humility." The second is the desire of things pertaining to knowledge, and this is moderated by "studiousness" which is opposed to curiosity. The third regards bodily movements and actions, which require to be done becomingly and honestly, whether we act seriously or in play. The fourth regards outward show, for instance in dress and the like" [ST II-II q160, a2].
Dress, external behavior, mannerisms, etc. are signs of the person, and become so in the cultural context in which the person lives, and in which it indicates something to others. The Christian conforms to the culture in such matters, unless sin is intrinsically involved (clothing which will have the general effect to tempt the opposite sex). Modesty is humility in dress and mannerisms, an outward sign of the disposition of the inner man. By not standing out the Christian assumes a humble posture toward his neighbors.
Whether men and women sit on opposite sides of the church, men wear a skull-cap, and women a veil, as the Jews of St. Paul's day did, is therefore ultimately a matter of modesty, and thus of custom. St. Paul even alludes to this in the Corinthians passage (v.16). When the "approved mores of the people" (1917 CIC, c1262, 2) change, the Church, desiring to be "all things to all men" (1 Cor. 9:22), can conform to those customs. Only the Magisterium is competent to determine which customs can legitimately be practiced, and where custom leaves off and divine law begins. We are always safe in following the Church, rather than our own judgment, for even if the Church makes a prudential error, it is "bound in heaven" (Mt. 16:13-18). A Sign of Subordination Even if wearing head-covering is not a moral obligation, isn't it a fitting sign of the subordination St. Paul speaks of in the passage in Corinthians? First, lets look at what subordination is. It means to be ordered (directed in an orderly way) toward a particular goal or end, sub (under) some other person's direction. A worker is subordinate to his supervisor, the supervisor to his manager, the manager to the owner, all in order that the company run smoothly to achieve its purpose. As persons, as citizens, as Christians, and in many other categories of existence, worker and supervisor are equals, but in working toward the goal of making the company's product they are not. Consider the examples St. Paul gives as to why women should be covered.
1 Cor. 11:3 But I want you to know that Christ is the head of every man, and a husband the head of his wife, and God the head of Christ.
1 Cor. 11:8-12 For man did not come from woman, but woman from man;  nor was man created for woman, but woman for man;  for this reason a woman should have a sign of authority on her head, because of the angels. 
In Christian marriage the husband is the head of his wife, as Christ is head of the Church. This is also St. Paul's message in Eph. 5:21-33, in which he enunciates the supernatural meaning of Christian marriage as a sacramental sign of Christ's union with the Church. St. Paul then goes on in Corinthians to recall the creation of man and woman, pointing out that woman was taken from man, not vice versa. As Pope John Paul II so clearly taught in his catechesis on Genesis, marriage is not only a Christian sacrament, it is a naturalsacrament of the Communion of Persons within the Trinity. What this tells us is that the equality of persons within a communion does not destroy the hierarchical order of the nature in which it exists. In the divine nature the Father is the head, in the Church it is Christ, and in marriage it is the husband. Indeed, in the Christian order the natural order is perfected, since love becomes, or should become, the motive force of all relations. No doubt this is why St. Paul, in his Ephesians discourse on marriage, begins it by saying, "defer to one another out of reverence for Christ." (Eph 5:21).
Why, then, would the Church drop the practice of such a fitting sign of the natural order? While, it is certainly still true that the husband has the headship in marriage and the family, I can think of several possible reasons.
1. Lost significance. As explained above, signs are culture specific. A particular gesture, clothing, expression, conveys a meaning which is widely understood by people of a particular culture. When the culture no longer sees the significance the sign loses its meaning, except to those who have retained the understanding of it. Certainly, the practice of an important sign can re-introduce a particular understanding into a culture, and so an argument can be made for retaining a sign, like women wearing a head covering in church, and teaching its significance. Indeed, this MUST be done in the case of the matter of the sacraments. Rice cakes cannot be used for the Eucharist, even where rice and not wheat is the staple food. The Church must simply teach the meaning of the sacramental sign. The wearing of a veil or other head covering is not a sign of that significance, however, and so when and where it has lost its meaning it can be set aside, as the Church has evidently done.
2. Conflict of meaning. A sign, while remaining valid, may nonetheless suggest a meaning that would be an obstacle for people in a particular culture. Take the case of white vestments. For Western Christians they convey joy and celebration, but in the Far East white connotes mourning and sadness. Should the Church hold onto her custom because of its longevity or conform it to the understanding of Oriental cultures? She chooses to make her liturgical signs understandable in the culture in which they must be "read." In the particular case of head covering, while the truth intended by this sign remains valid, properly understood and in union with other truths, it is easily misconstrued today as a servile subordination of wife to husband or even all women to all men. In the contemporary world, in which the equality of men and women as persons is emphasized, this is a legitimate consideration. We must not use our Christian freedom to hinder souls (1 Cor. 8). Since there is no intrinsic moral obligation to this practice, it can be set aside. As the last canon of the Code of Canon law reminds us, the salvation of souls is the highest law of the Church (salus animarum suprema lex).
3. Liturgical theology. Among the doctrinal truths manifested in the Mass is the hierarchical nature of the Church. The Church, the Mystical Body, is composed of Christ the Head and those who have been baptized into Christ, His members. The visible distinction of offices in the Liturgy, between the ministerial priesthood on one hand and the people on the other, are the sacramental sign of the Mystical Christ, Head and members. Within that liturgical, sacramental order, except for the fact that those who represent Christ the Head must be male, the natural distinction between the sexes and within marriage is not liturgically significant. In baptism "there is no longer male or female" (Gal. 3:28). Thus, 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. Personal Piety While it is absolutely clear to me that there is no canonical or moral obligation for women to wear a head-covering in Church, women are certainly free to do so as a matter of personal devotion. They should, however, see it as a sign of subordination to God, as that better suits the liturgical context. Those who wear a covering or veil, and those who don't, should not judge the motives of the other, but leave each woman free in a matter that is clearly not of obligation.
Answered by Colin B. Donovan, STLrevised 28 April 2011