The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$. 


I tried to generalized this concept to a higher dimension, and consider the quotient of $(S^1)^3$ by the action of the symmetric group $S\_3$ on the coordinates. 


I was able to compute the homology of this space: $H\_n = \mathbb{Z}$ for $n = 0,1$, and 0 otherwise (with reduced homology being 0 at $n=0$ as well).


Even with this information I wasn't able to identify said space in any other way. Is it well known, or, can it be described in any other fashion? What can be said about higher dimensions?



Quotient of Three Dimensional Torus by Permutation on Coordinates
What you are describing are the *symmetric powers* of $S^1$. The symmetric power $SP^n(S^1)$ is a fibre bundle over $S^1$ with fibre an $(n-1)$-simplex. (In particular your calculation of the homology for $n=3$ checks out.) This bundle is orientable if $n$ is odd, and non-orientable if $n$ is even.


This result is due to H. R. Morton, and is nicely written up at the nLab page <https://ncatlab.org/nlab/show/symmetric+product+of+circles> 


