Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact? It is well known the existence of a T duality between the two heterotic string theories, $SO(32) \sim_T E_8 \times E_8$. Beyond the trivial point that both groups have the same dimension (496, which actually is a prerequisite), is there some other mathematical relation between them?
I am thinking in other SO(N) groups whose dimension is a perfect number and that happen to be related to products of manifolds. $SO(4)$ with $SU(2) \times SU(2)$, and -I am told- $SO(8)$ with some variant of $(S^7 \times S^7) \times G_2$. It should be nice if all of these were justified by a common construction, but I am happy just with an answer to the $SO(32)$ case.
 A: The answer to this question can be found in Lubos Motl's answer to this question of mine on Physics.SE. 
The key here are the weight lattices bosonic representations $\Gamma$ of these gauge groups.
As I understand it, the weight lattice of $E(8)$ is $\Gamma^8$, whereas the weight lattice of $\frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2}$^ is $\Gamma^{16}$. The first fact means that the weight lattice of $E(8)\times E(8)$ is $\Gamma^{8}\oplus\Gamma^8$,    
Now, an identity, that $\Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}=\Gamma^{16}\oplus\Gamma^{1,1} $ , which actually allows this T-Duality. Now, this means that it is this very identity which allows the identity mentioned in the original post. 
So, the answer to your question is "Yes", there is a group-theoretical fact, and that is that   $ \Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16}\oplus\Gamma^{1,1} $.         
A: It's worth noting this fact.  Suppose $L$, $L'$ are even unimodular Euclidean lattices, like $\mathrm{E}_8= \Gamma^8$ or $\Gamma^{16}$ or any direct sum of these, but also the Leech lattice and 23 others in 24 dimensions, and at least 80,000,000,000,000,000 others in 32 dimensions, etc.  Let $\Gamma^{1,1}$ be the even unimodular Lorentzian lattice in 2 dimensions.  Then
$$L \oplus \Gamma^{1,1} \cong L' \oplus \Gamma^{1,1}$$   
The reason is that there's only one even unimodular Lorentzian lattice in dimensions $8n + 2$, up to isomorphism (and none in other dimensions).
So, the fact that $\mathrm{E}^8 \oplus \mathrm{E}^8$ and $\Gamma^{16}$ become isomorphic when you take their direct sum with $\Gamma^{1,1}$ is an instance of this general pattern.
