Princess Margarete of Prussia, daughter of Victoria, the Princess Royal (and granddaughter of Queen Victoria) gave birth to TWO sets of twins. She already had two sons from single births before having the two sets of twins in 1896 (Philipp and Wolfgang) and 1901 (Richard and Christoph). When the Queen heard about the first set of twins, she was said to have laughed about the sudden increase in great-grandchildren.
At the end of WW1, Finland wanted to establish a monarchy, so they elected Margarete's husband, Prince Friedrich Karl as king. Their two older sons had both been killed in WW1, so they only had their two sets of twins left. Wolfgang, the younger twin of the first set, was chosen as his father's heir to the Finnish throne, since he was at the time ready to go with his parents to Finland, while his elder twin brother was not. Under pressure from the Allies, who wouldn't accept a German prince as king of Finland, Friedrich Karl renounced the throne, and Finland abandoned the idea.
I know quite of a few of Queen Victoria's descendants had twins, but this is the only case I can think of where a set of twins had a more active role in the politics of a country, though in this case, it was the younger twin who was preferred over the older one!