This doesn't necessarily apply to the diaper-on-the-head coverings JW women use when there are no men present, but....
I wear a veil to Mass. Here is what Dr. Alice von Hildebrand has to say about it:
"...there is something extraordinarily great and mysterious about femininity. And why do I say it is so great and so mysterious? Because you all know that every little girl that is born, is born with a seal, so to speak, protecting the mystery of her femininity, which is the womb. There is a seal and if you understand, a seal always indicates something which is sacred. The seal, which doesn’t exist in the male body, is profoundly symbolic and says this belongs to God in a special way. This is a sphere which is so beautiful and so profound that it cannot be touched upon, except with God’s permission, in a Catholic marriage.
When a girl or young woman is permitted to give the keys of this mysterious domain, this closed garden, to her husband-to-be, she says: "Up until now I have kept this garden virginal, now God has given me the keys and is allowing me to give them to you and I know that you will penetrate into it, with trembling reverence and gratitude". The moment that a woman is embraced by her husband and a few hours afterwards she conceives, in this very moment, something absolutely amazing happens which once again illuminates the greatness of femininity. Neither husband nor wife can create a human soul. God alone can.
Of course there is the male seed and there is the female egg. These are material realities that God has put into the bodies and when they are united, an amazing thing happens. God creates a new human soul, totally new, which never existed before. Where? In the mystery of the female body. This is where the soul is conceived. It has nothing to do with the husband. The husband is out of the game at this point and the very moment that God creates a soul he implies that there is a special contact between God and the female body, so to speak, touching it in creating it. Once again, what an extraordinary privilege.
And this is why the female body should be veiled because everything which is sacred calls for veiling. When Moses came down form Mount Sinai, he veiled his face. Why did he veil his face? Because he had spoken to God and at that very moment there was a sacredness that called for veiling.
Now the stupid feminists after Vatican II suddenly "discovered" that when women go to Church veiled, it is a sign of their inferiority. The man takes off his hat and the woman puts on a veil. My goodness, how they have lost the sense of the supernatural. Veiling indicates sacredness and it is a special privilege of the woman that she enters church veiled.
You see the Church recognises things so profoundly that in some way you can say she has always recognised the special dignity granted to women. You cannot be a Christian and not recognise the privilege that it is to be a woman, because the most perfect of all creatures, the only creature born without original sin, is a woman and therefore once again you understand the extraordinary privilege of being one and having this image of the Holy Virgin, who was both Virgin and Mother and the two go beautifully together."