To get a little background for this scripture, start with 9:19-23.

I Cor. 9 (19) For though I be free from all men, yet have I made myself servant unto all, that I might gain the more. (20) And unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law [Torah], as under the law [Torah], that I might gain them that are under the law [Torah]; (21) To them [the Gentiles] that are without law [Torah], as without law [Torah], ([Myself] being not without law [Torah] to YAHWEH, but under the law [Torah (as mediated by the Messiah under the terms of the New Covenant and not the Older Covenant)] to Messiah,) that I might gain them [the Gentiles] that are without law [Torah]. (22) To the weak became I as weak, that I might gain the weak: I am made all things to all men, that I might by all means save some. (23) And this I do for the gospel's sake, that I might be partaker thereof with you.

I think that Paul was telling the Assembly at Corinth that for the Gospel’s sake he did everything he could to be sensitive to racial, social and cultural patterns of all people. In other words, he put himself in their shoes and intermingled with them. He even participated in such things as cleansing rituals which were administered under the Levitical Priesthood. He did these things with them in order to be able to teach them Messiah. Even James and the Elders of the Assembly at Jerusalem advised Paul to do the same.

Acts 21 (20) And when they [James and all the elders] heard it, they glorified YAHWEH, and said unto him, Thou seest, Brother, how many thousands of Jews there are which believe; and they are all zealous of the law [Torah]: (21) And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs [of the Torah]. … (23) Do therefore this that we say to thee: We have four men which have a vow on them; (24) Them take, and purify thyself with them, and be at charges with them, that they may shave their heads: and all may know that those things, whereof they were informed concerning thee, are nothing; but that thou thyself also walkest orderly, and keepest the law [Torah]. … (26) Then Paul took the men, and the next day purifying himself with them entered into the temple, to signify the accomplishment of the days of purification, until that an offering should be offered for every one of them.

Also, for the sake of the Gospel Paul circumcised Timothy.

Acts 16 (I) Then came he to Derbe and Lystra and, behold, a certain disciple was there, named Timotheus, the son of a certain woman, which was a Jewess, and believed; but his father was a Greek: (2) Which was well reported of by the brethren that were at Lystra and Iconium. (3) Him would Paul have to go forth with him; and took and circumcised him because of the Jews which were in those quarters: for they knew all that his father was a Greek.

Did Paul, James and the Elders of the Assembly at Jerusalem conspire together in a lie to show that Paul was keeping the Torah himself and to show the Believing Jews that he was not teaching the Jews among the Gentiles to forsake the teachings of Moses by saying that they ought not to circumcise their children, neither to walk after the customs [of the Torah]? Or, were they (without bringing themselves under the bondage of the Torah) upholding these Torah traditions and customs in the letter as a teaching tool which the Torah of the Old Covenant was for in the first place?

Gal. 3 (21) Is the law then against the promises of YAHWEH? YAHWEH forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law. (22) But the scripture hath concluded all under sin, that the promise by faith of YAHshua Messiah might be given to them that believe. (23) But before faith came, we were kept under the law, shut up unto the faith which should afterwards be revealed. (24) Wherefore the law [Torah] was our schoolmaster to bring us unto Messiah, that we might be justified by faith. (25) But after that faith is come, we are no longer under a schoolmaster.

Surely at the same time that Paul, James and the Elders of the Assembly at Jerusalem participated in these things, they explained to the their Jewish brothers who were still in bondage under the Torah what those Torah traditions or customs signified. It was in this manner that they were taught Messiah. After they were won over for Messiah, they were surely taught the Torah as mediated by the Messiah under the terms of the New Covenant which replaces the Torah of older covenants.

Therefore, Paul, James and the Elders of the Assembly at Jerusalem all were very sensitive to the racial, social and cultural patterns of all people. Otherwise, they would not have been able to lead their brethren unto Messiah.

I think that there were many of those Torah traditions that were no longer required, but Paul went along with them that he may be able to teach them Messiah. He also went along with other ordinances or traditions as long as they did not violate YAHWEH'S Torah. Consequently, Paul passed these traditions down to the Believers as he wanted them to be sensitive to racial, social and cultural patterns of all people too.

Without a doubt, he knew what all of their traditions were for he admits that he did.

Gal. 1 (14) And [I, Paul] profited in the Jews' religion above many my equals in mine own nation, being more exceedingly zealous of the traditions of my fathers.

Keep the above things in mind.

I Cor. 11 (1) Be ye followers [imitators] of me, even as I also am of Messiah.

In order for the Believers to be able to assemble in the synagogues with the Jews where they had their best chance to teach them Messiah, they needed to imitate Paul as well as James and the Elders of the Assembly at Jerusalem as they are all of Messiah. The Believers also had to be very sensitive to the Jewish traditions or ordinances while in their Temple and in their synagogues. (The synagogues were the most common place for Believers to assemble for many years after the Temple was destroyed until they were literally forced out.)

Paul passed on all of those traditions to the Assembly at Corinth.

I Cor. 11 (2) Now I praise you, brethren, that ye remember me in all things, and keep the ordinances [traditions G3860], as I delivered them to you.

Strongs G3860; transmission, i.e. (concerning.) a precept; specifically the Jewish traditionary law:--ordinance, tradition.

NASC G3862. paradosis, from G3860; a handing down or over, a tradition:-- tradition, traditions.

The headship tradition mentioned in 11:3 has always been part of Jewish thinking and tradition that Paul is passing on here. (The Jews still think this is the correct order, but they do not believe that YAHshua is the one Who was to be their Messiah.)

I Cor. 11 (3) But I would have you know, that the head of every man is Messiah; and the head of the woman is the man; and the head of Messiah is YAHWEH.

The veiling of the head and face tradition mentioned in I Cor. 11:4-11 was Jewish thinking and tradition as the culture of all Eastern countries were and still are for the women to veil themselves in the presence of all men except their own husbands. This is especially true in the presence of holy men. (Verse three is part of the context, so do not disconnect it from the following scriptures)

I Cor. 11 (4) Every man praying or prophesying [in public worship], having his head covered [i.e. literally pulling something down over the head], dishonoureth his head [i.e. Messiah]. (5) But every woman that prayeth or prophesieth [in public worship] with her head uncovered dishonoureth her head [i.e. man]: for that is even all one as if she were shaven. (6) 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. (7) For a man indeed ought not to cover his head [in public worship], forasmuch as he is the image and glory of YAHWEH: but the woman is the glory of the man. (8) For the man is not of the woman; but the woman of the man. (9) Neither was the man created for the woman; but the woman for the man. (10) For this cause ought the woman to have [submissive] power on her head because of the angels [Holy men were also referred to as "angels"]. (11) Nevertheless neither is the man without the woman, neither the woman without the man, in the Sovereign. (12) For as the woman is of the man, even so is the man also by the woman; but all things of YAHWEH.

While imitating Paul, James and the Elders of the Assembly at Jerusalem for the gospel’s sake, and while worshiping in the synagogues in the time of Paul, the answer to verse thirteen was: "No," it would not be comely in that culture; to verse fourteen, "Yes."

I Cor. 11 (13) Judge in yourselves: is it comely that a woman pray unto YAHWEH uncovered? (14) Doth not even nature itself [the nature of things i.e. a man should look like a man and a woman should look like a woman] teach you, that, if a man have long hair, it is a shame unto him? (15) But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.

It seems that Paul was telling the New Covenant Believers of his day to imitate him while they are attending public worship which was mainly in the Jewish synagogues by being sensitive to their traditions and ordinances for the sake of the Gospel, but in their private worship those traditions and ordinances were not mandatory or necessary.

I Cor. 11 (16) But if any man seem to be contentious, we have no such ordnance [or tradition concerning whither to pray with or without shawls covering the head], neither the Assemblies of YAHWEH.

I think that with very few possible exceptions, the veil is not necessary today.

Gal. 3 (26) For ye are all the children of YAHWEH by faith in Messiah YAHshua. (27) For as many of you as have been baptized into Messiah have put on Messiah. (28) There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Messiah YAHshua. (29) And if ye be Messiah's, then are ye Abraham's seed, and heirs according to the promise. [HALLELUYAH!]