A space with "interchangeable" coordinates, $\mathbb{R}^n / S_n $ (I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best):
I have a space that is similar to $\mathbb R^n$ with the following property: the point $(x_1,x_2,\ldots,x_n)$ is identified with any other point that has the same coordinates in permuted order.  For concreteness, if my space is based on $\mathbb R^2$, then the point $(0,1)$ would be identified with the point $(1,0)$ and would be indistinguishable from it.
From what I can glean, this space is $\mathbb R^n$ with a group action associated with it, where that group action is due to the permutation group permuting the coordinates.  Is that all it is?  Does it have any nice properties (for instance, is it a manifold)?  How would I begin to study this space?
 A: It sounds like you're describing the quotient space $\mathbb{R}^n/S_n$. Some people call this the $n^{th}$ "symmetric power" (of $\mathbb{R}$), although be a little careful with that terminology because it can be used to refer to two other related but different constructions. 
This quotient is not a manifold, but can be thought of as an orbifold. For example, $\mathbb{R}^2/S_2$ is given by folding the plane in half along the diagonal $x = y$, and so fails to be a manifold (without boundary) on the diagonal, where the action of $S_2$ has nontrivial stabilizer. 
If you toss out every point where the action of $S_n$ has nontrivial stabilizer (so, every point where some two of the $x_i$ are the same) you get a space called the configuration space of $n$ unordered points in $\mathbb{R}$. This is not a very interesting space, but its relatives (e.g. the configuration space of $n$ points in $\mathbb{R}^k$ for $k \ge 2$) are very interesting and extensively studied. 
There is a related construction in algebraic geometry that is somewhat better behaved and also called the symmetric power. 
