Traditionally (well according to some traditions), the decorations come down on the twelfth day of Christmas....6th January. In Christianity, it is Epiphany - the day day when allegedly the three Kings ('wise men' visited the baby Jesus, and his glory was realised (epiphany = a striking realisation).
In Italy the day is called Befana, a variation on 'Epiphany'. A broomstick-riding old woman, the Befana, brings gifts to children, along with a lump of "coal" - black sweets - for the times they have not been good during the year.
The legend told of her is that, having missed her opportunity to bring a gift to the child Jesus together with the three kings, she now brings gifts to other children on that night.
Go Befana, I say! Trust the blokes to cut the women out of the action.
In some Christian sects, eg the Eastern Orthodox, Epiphany is considered the most important day, and gifts are exchanged then, rather than on Christmas Day.
January 5th , the Twelfth Night, the night before Epiphany, was often celebrated with feasting in England in previous centuries, probably partly inspired by Shakespeare's play.