These are more "concepts" than symbols, but boy oh boy are they important.
In fact, the biggest concepts you're going to find in Common Sense are the concepts of independence and freedom. For starters, Thomas Paine is certain that independence and good government are the only things that will keep a country from falling into civil war.
As Paine writes,
… the most powerful of all arguments, is, that nothing but independence, i.e. a continental form of government, can keep the peace of the continent and preserve it inviolate from civil wars. (3.38)
Basically, it's better for America to go to war with Britain to get its independence instead of going into civil war with itself later.
For Paine, there is no doubt that as Americans, "A government of our own is our natural right" (3.50). He insists that any country whose rulers aren't democratically elected is not a free country at all. And it's only by having elected officials that a country can start
… securing freedom and property to all men, and above all things, the free exercise of religion, according to the dictates of conscience. (3.47)
Some of this stuff might look recognizable, because many of these same ideas ended up forming the basis of the United States Constitution.