The scriptural basis for women covering the head in church is a single passage, the eleventh chapter of Paul's first letter to the Corinthians, and it is among the most argued over in the New Testament. It instructs that a woman praying or prophesying with her head uncovered dishonours her head, and grounds this in an argument about headship, angels, hair and nature that has never been read the same way twice. Some interpreters take it as a universal ordinance; others as addressing a specific local situation in Corinth; others read the passage as concerned with hair worn up or loose rather than with a cloth at all. The passage also, incidentally, assumes women praying and prophesying aloud in the assembly, which sits awkwardly with the use later made of it.
Whatever the exegesis, the practice became general, and the 1917 Code of Canon Law made it a rule: women were to have their heads covered in church, particularly when approaching communion.
The 1983 Code simply did not repeat the provision. It was not abolished by an act of abolition; it lapsed by omission, which is why arguments about whether the obligation persists were possible at all. In ordinary parish practice the veil disappeared within a decade, and its recent return among some younger Catholics is a deliberate choice rather than an inherited habit, and is read as such by everyone in the building.
One curiosity belongs to the mantilla specifically. A small number of Catholic queens hold the privilege du blanc, the right to wear white rather than black in audience with the pope, where all other women wear black with a black veil. The list of those entitled is very short.
In Spain the mantilla is worn over a peineta, a high comb.