At the wedding of Prince Friso, his bride Mabel suddenly appeared with a sun-ray motif diamond diadem. It was told that the diadem was a second monture from the diadem belonging to the Mellerio Parure of diamonds and sapphires.
This means that there is probably a cassette or a box with more étages in which various montures (also for aigrettes and broches) are displayed, with a screwdriver next to it.
But indeed, thinking about Queen Elizabeth's diadems: the Girls of GB&I, the fringe, the Burmese Rubies, the Aquamarine diadem, etc. I do not see the need for screwdriving. Maybe it is more for broches, bracelets and colliers.