(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?