Grundlovsdag – Constitution Day (Denmark)

Constitution Day (Danish: Grundlovsdag), observed on June 5, commemorates the anniversary of the signing of the Danish constitution of 1849, which established Denmark as a constitutional monarchy, and honors the constitution of 1953, which was adopted on the same date Almost all workplaces and shops are closed at noon on this particular day and political rallies and meetings are traditionally held.

The day, however, is not widely used to celebrate the constitution or national day as is seen in other countries, such as in Norway.

It is a day of the people, and picnic baskets are packed for families and friends to venture out into the hopefully not rainy Danish early summer landscape where they will enjoy the scenery, while listening to speeches in celebration of democracy.
Denmark doesn't have a national day, so this day comes the closest to being that, and from 1891-1975 it was indeed a holiday, where people enjoyed a 1/2 day off. This is no longer the case by law, thus it is up to the individual company to decide their policy on the issue; but most shops are closed on this day.
Incidentally June 5th is also Father's Day, "Far's Dag", in Denmark.