TITLE: What is symmetric about the symmetric group?
QUESTION [16 upvotes]: In my abstract algebra reader many times they use the word "symmetry". 
But I just found out that I'm not quite sure what they mean with it.

Let $X=\{1,...,n\}$. The set $S_n$ of bijections $X\to X$ under
  composition with identity $X\to X :x\mapsto x$  is called the
  symmetric group on $n$ symbols.

If I think about symmetry I think about figures, as something I can visualize. However I'm not sure if or how I should visualize the symmetric group of $n$ symbols as something symmetric. 
Edit: Thanks for all the answers! They all contributed to be finally satisfied with calling $S_n$ symmetric :)
Edit2: This helped me a lot as well: http://en.wikipedia.org/wiki/Frucht%27s_theorem

Frucht's theorem says that every finite group is the symmetry group of
  some graph. So every finite abstract group is actually the symmetries
  of some explicit object.

REPLY [2 votes]: The origin of symmetric group began with geometric idea. Given for example equilateral triangle $ABC$, we search all rotations and reflexions which leave invariant the triangle, so the rotations and reflexions which send a vertex to another. 
If we denote $A=1$, $B=2$ and $C=3$, a possible rotation is that send $1$ to $2$ and $2$ to $3$ and $3$ to $1$ hence we find the permutation:
$$\sigma=\left(\begin{array}{cc}1&2&3\\2&3&1\end{array}\right).$$