Look at them from a distance and you might wonder if they might be related. Yet Madonna, the American singer, shares no bond with Marine Le Pen, the scion of a French political dynasty and the leader of the French National Front. Now, after a number of hostile comments between them, the pair have said they would discuss their differences over a drink. Can a modern pop icon change the way Le Pen thinks? Or will the face of Europe's new far-right populism win over Madonna?

The unlikely dispute between the pair goes back years. In 2012, during her MDNA world tour, Madonna began showing a video of Le Pen's face in her live show. Le Pen wasn't the only international name targeted – Sarah Palin and Hu Jintao were among the others shown – yet the appearance of Le Pen was especially notable due to a brief portrayal of a swastika on Le Pen's forehead. To hammer home the point, the image comes just before another of Adolf Hitler.

The swastika was clearly a sore point for Le Pen. The National Front is now seen as the third-largest party in French politics, verging on mainstream with a new focus on immigration. But when it was founded by her father, Jean-Marie Le Pen, in 1972, the party had greater links to the far-right fringes: Jean-Marie himself was accused and convicted a number of times of xenophobia and anti-Semitism. Since Marine Le Pen took power in 2011, the National Front has struggled to avoid the "Nazi" label – an especially tough tag to have in a country that was occupied by the Germans during World War II.

Le Pen's response to the image was swift. "It's understandable when aging singers who need publicity go to such extremes," she told the press, warning Madonna that if her image was shown in this way when the MDNA tour came to France, she would sue. After the American singer used the image in a show in Paris, the National Front leader made good on her promise, filing a lawsuit for "public insult." Madonna subsequently switched the swastika for a question mark.

Madonna was accused of "appeasement" in the French press, and at her shows she addressed the controversy, saying it was not her "intention to make enemies" and that she wanted to "promote tolerance." She may have been taken aback by some of the negative reaction in France to her image of Le Pen – even Najat Vallaud-Belkacem, a spokeswoman for France's Socialist government, said the image of Le Pen and a swastika was "unfortunate." Ticket sales for the French leg of the MDMA tour were said to be poor.

Flash forward to this year, and Madonna has reignited the feud. Last week, while talking on France's Europe-1 radio show, she warned of a rise of far-right attitudes in Europe and described the National Front as "fascist." Comparing the situation across the continent to Nazi Germany, she said that France was particularly bad. “The level of intolerance is so enormous it’s scary," Madonna said, erroneously saying she had been receiving threats from "Marie Le Pen."

The situation looked like a redux of 2012, with the National Front again announcing its intention to legally challenge Madonna when she came to France later that year on tour. Then, things took a strange turn. On Monday, Madonna told Canal+ that she might have "misunderstood" Le Pen and said she'd like to "sit down and have a drink" with the National Front leader.

Le Pen initially refused, but she changed her mind Tuesday. "I accept Madonna's invitation with pleasure," she told AFP. "I appreciate people who make overtures in good faith." It seems like some kind of alcohol-related summit may really be in the cards between the two bitter enemies: The "Material Girl" and "The Devil's Daughter" will sit down over some wine, perhaps, and discuss human rights.

Le Pen's decision to accept the invitation has the effect of making her look like the mature one of the two. Perhaps most importantly, it shows her as secure. That's understandable: Madonna's creative and commercial peak was in the late 1980s and 1990s. In a post-Paris shooting world, Le Pen sees her political peak ahead of her.

Polls suggest she could be right.