During the monarchy, liberals were left wing (the "radical republicans"). After they won, liberals became the defenders of the status quo and are thus the right wing. There are people further right that wish to roll the clock backwards.
The Radical Republicans refers to a group of anti-slavery congressmen in the 1800s up to, during, and after the American Civil War. There was no monarchy at the time.
That was one later instantiation. Republican radicalism was feared by the aristocracy in the 1700s-1800s. Perhaps they weren't uniformly called "radical" but republicanism was hated.
The group you're referring to was part of the "Republican Party" which was created after republicanism was victorious in the united states after the revolution. I'm not intimately familiar with the iconography around that period, but I would speculate that they were drawing on the old symbols of radicalism when creating that name.
