Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H (if G is connected)." In mathematics, duality is a very important phenomenon and one may immediately come up with lots of examples(e.g., dual space in functional analysis). At the same times, there are many different kinds of symmetry in mathematics. This wiki article points out that "A high-level concept related to symmetry is mathematical duality.".
Here is my question:
What's the difference and relationship between "duality" and "symmetry"?