I have finally worked out how to do it.
You turn the duvet cover inside out and lay it out on the bed. Lay the duvet on top of it. Take the corners of the duvet and poke them inside so they match up to the corners of the cover. Then slowly roll inside out, making sure the duvet is straight. You'll do it in about a minute.
I've also learned how to slice bread. Both skills only took about 40 years to learn.