The difference between men's and women's clothing was small but distinctive. In addition, men often wore a turban to confine their hair, and women of some cultures wore a veil.
A woman must not wear men's clothing, nor a man wear women's clothing, for the LORD your God detests anyone who does this. (NIV, Deuteronomy 22:5)
No one knows for sure whether this prohibition was intended as a general principle or was directed at some specific abuse among the ancient Hebrews. Cross-dressing was likely considered an affront to the natural distinction between the sexes (Genesis 1:27). It may also have been related to some deviant sexual practice, or more likely, to pagan worship. It is known that some pagan rituals of that time involved women wearing armor and men dressing as women, and the Hebrews were forbidden to do anything that had even the appearance of pagan worship.
Some people think this verse would prohibit women from wearing pants because pants have traditionally been worn by men. But, in light of the similarity of men's and women's clothing in Biblical times and the fact that pants were not worn by either sex at that time, that conclusion would be difficult to justify.
Do not wear clothes of wool and linen woven together. (NIV, Deuteronomy 22:11)
Again, no one is sure why this would be wrong. It may have been to avoid mixing things that God has created separately (Deuteronomy 22:9-11, Leviticus 19:19). Or, these mixtures may have been related to some idolatrous practice that the Hebrews were forbidden to imitate.
Related article: What Does the Bible Say About the Old Testament Law?
I also want women to dress modestly, with decency and propriety, not with braided hair or gold or pearls or expensive clothes, but with good deeds, appropriate for women who profess to worship God. (NIV, 1 Timothy 2:9-10)
Your beauty should not come from outward adornment, such as braided hair and the wearing of gold jewelry and fine clothes. Instead, it should be that of your inner self, the unfading beauty of a gentle and quiet spirit, which is of great worth in God's sight. For this is the way the holy women of the past who put their hope in God used to make themselves beautiful. (NIV, 1 Peter 3:2-5)
Many of the New Testament letters address specific abuses that occurred in the early Christian communities, and that may be the case here. Jesus had defied the standards of first-century male-dominated society by treating women as equals. Paul had declared all people equal in the family of God (Galatians 3:26-29). Many Bible scholars believe some women in the church had carried their "liberation" too far and adopted offensive styles of dress. Church leaders were anxious to avoid any hint of scandal in the churches, and these passages served that purpose.
Both of these passages also make the point that a person's true beauty comes from within and is properly expressed by good deeds rather than showy clothing and jewelry.
A few Christians interpret these passages as a requiring women to dress very plainly and refrain from wearing jewelry or using makeup. But most Christians believe the advice is simply to dress modestly and in good taste, according to the standards of the society they live in.
Any man who prays or prophesies with something on his head disgraces his head, but any woman who prays or prophesies with her head unveiled disgraces her head--it is one and the same thing as having her head shaved. For if a woman will not veil herself, then she should cut off her hair; but if it is disgraceful for a woman to have her hair cut off or to be shaved, she should wear a veil. For 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. (NRSV, 1 Corinthians 11:4-7)
In first century culture, it was considered a mark of respect for a man to remove his turban in the presence of a superior. Similarly, a man should remove his head covering when he came into God's presence in prayer. Additionally, it was the custom of pagan men to cover themselves while praying, so as to avoid distractions. Thus, men should remove their head coverings to avoid any association with paganism.
Jewish women did not normally wear veils, but reputable Greek and Roman women did. A woman's veil was a symbol of her modesty and respect for her husband. For a Corinthian woman to remove her veil in public would have been an insult to her husband and an affront to the Greek/Roman society in which she lived. Paul strongly discouraged any such rebellion or hint of scandal within the churches. In addition, some pagan priestesses removed their veils and wore their hair disheveled when prophesying. Thus, women should remain veiled while praying or prophesying to avoid any association with paganism.
It must have been a great disgrace for a woman to shave her head. So, Paul made the comparison that removing one's veil while praying or prophesying would be an equally great disgrace.
1 Herbert Lockyer, Sr., ed., Nelson's Illustrated Bible Dictionary, Thomas
Nelson Publishers, 1986.
2 Merrill F. Unger, The New Unger's Bible Dictionary, Moody Press, 1988.