The symmetric square of a sphere $\DeclareMathOperator\Sym{Sym}$Recently, I have been thinking about the space $\Sym^2(S^n)$ of pairs of points in the sphere (including the repeated pairs $(p,p)$ for each $p\in S^n$) and possible ways to describe it. This is topologized as the quotient of $S^n\times S^n$ under the involution that switches components. I am looking (somewhat broadly and ill-definedly) for descriptions of this space and related results. I will describe below some of the results that I have encountered and I would really appreciate any pointers towards descriptions that I have missed in my literature search.
(Side note: If the case of $n=1$ is something you have thought about / want to think about / enjoy, also consider taking a look at this related question that I asked on StackExchange about different ways to prove that $\Sym^2(S^1)$ is a Möbius strip.)
Primarily, I am interested in the homeomorphism type of $\Sym^2(S^n)$, of which I have found two descriptions:

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*On the topology of cyclic products of spheres by Liao cites a description due to Steenrod, where $\Sym^2(S^n)$ is formed by attaching a $2n$-cell to the (unreduced) suspension of $\Sym^{2}(S^{n-1})$. This seems quite useful, since it gives an inductive view of $\Sym^2(S^n)$ as a CW-complex.

*On the symmetric square of a sphere by James, Thomas, Toda and Whitehead gives a second description, as the mapping cone of a certain map $\Sigma^n(\mathbb R\mathbb P^{n-1})\rightarrow S^n$ (here $\Sigma$ denotes the unreduced suspension). This seems more specialized than the above (their calculations actually use both descriptions), but it also seems to be the more-cited description (perhaps this is because it appears in Hatcher's Algebraic Topology, on page 482).

These are the only two ways I know of to describe the homeomorphism type and I would be very interested if there are others that I have missed.
In terms of computations, it seems that a lot is known about the homology, homotopy and $K$-theory of the space $X_n=\Sym^2(S^n)$. However, these results are widely scattered throughout the literature, so there's a very reasonable chance that I have missed things. Here is what I have been able to find:

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*It seems that the homology groups $H_i(X_n;\mathbb Z)$ are well-understood and should follow from Dold's purely algebraic recipe in Homology of symmetric products and other functors of complexes (though it was known before this too). I have no idea what is known about the cup product for this space.

*In Homotopy of two-fold symmetric products of spheres, Nakaoka describes the stable homotopy groups $\pi_{n+i}(X_n)$ for $0<i\leq \min(2n-2,9)$. For $n>1$, the space $X_n$ is simply-connected, so we also get $\pi_n(X_n)=\mathbb Z$ and $\pi_i(X_n)=0$ for $0<i<n$, by the Hurewicz theorem. I don't know if any other homotopy groups of these spaces are known.

*For the real $K$-theory of $X_n$, we have $K^{n+1}(X_n)=0$ and the inclusion $S^n\rightarrow X_n$ induces an injection $K^n(X_n)\rightarrow K^n(S^n)=\mathbb Z$ of index $2^{k}$, where $k$ is the number of integers $0<s<n$ such that $s\cong 1,2\text{ or }4\bmod8$. (See the James–Thomas–Toda–Whitehead paper mentioned above.)

And that's all I know about these spaces! I would be very happy to hear any details that I have missed here.
 A: That cofiber description in the old short paper of James and other famous folks tells you a lot.
The map you have turns out to be adjoint to the standard map $\mathbb RP^{n-1} \rightarrow \Omega^n S^n$.
Continuing the cofibration sequence one place to the right gives a cofibration sequence
$$ S^n \rightarrow SP^2(S^n) \rightarrow \Sigma^{n+1}\mathbb RP^{n-1}$$
that ends up being short exact in (co)homology with $\mathbb Z/2$ coefficients.
Furthermore, all possible Steenrod operations acting on the class in dimension $n$ are nonzero: with $x_k$ denoting the nonzero class in degree $k$, for $k=n$ or $n+2\leq k \leq 2n$, one has  $Sq^i(x_n) = x_{n+i}$ for $i=2, \dots, n$.  This tells you that there is one interesting cup product: $x_n^2 = x_{2n}$.
The $K$-theory can be worked out with the Atiyah-Hirzebruch spectral sequences.
More fun happens when $n=\infty$, so we are looking at $SP^2(S)$, where $S$ is the sphere spectrum.  Now one has a cofibration sequence of spectra
$$ \Sigma^{-1} SP^2(S) \rightarrow \Sigma^{\infty} \mathbb R P^{\infty} \rightarrow S$$
that, quite remarkably, induces a short exact sequence in homotopy groups in positive degrees: this is a consequence of the Kahn-Priddy Theorem.  This also is short exact in all Morava K-theories. Fun, fun, fun!
