If one thinks of $SO(3)$ as the rotations in three-dimensional space, then the $\mathbb Z_2\times \mathbb Z_2$ subgroup is given by the 180 degree rotations along three perpendicular axes (which indeed commute).
On a mathematical level this just seems like --at least on first sight-- one of the many finite subgroups of $SO(3)$. However, I am a physicist, and there this subgroup plays an important role. In particular, physicists are interested in the fact that $H^2_\textrm{group}(SO(3);U(1)) \cong H^2_\textrm{group}(\mathbb Z_2 \times \mathbb Z_2; U(1))$, i.e. if one is interested in the distinct classes of projective representations of $SO(3)$, it is sufficient to focus on $\mathbb Z_2\times \mathbb Z_2 \subset SO(3)$. (This ties into something physicists call topological phases of matter; in fact this particular piece of math is relevant for the recent Nobel prize.) So at least from this perspective, $\mathbb Z_2\times \mathbb Z_2$ can be seen as a sort of `skeleton' of $SO(3)$ (in purely figurative language).
So I was wondering: is there something mathematically significant about the $\mathbb Z_2 \times \mathbb Z_2$ subgroup of $SO(3)$? I.e. is there some natural characterization which defines it as a subgroup, leading to a notion that is applicable to more general groups? (Or is the fact that it is has the same [second] group cohomology simply an uninteresting and limited curiosity?) Is its relation to the generators of $SO(3)$ of particular importance?
You can get a $Z_2\times Z_2$ embedded in $SO(3)$ also via the Klein subgroup of $S_4$ and observe that $S_4$ is the group of symmetries of the regular tetrahedron.
In the book Spin Geometry by Lawson and Michelsohn, on pages 36-37 the authors discuss the finite Clifford group $F_n$ associated with the Clifford algebra $Cl(n)$, as well as the relation between their representations. Since Clifford algebras and matrix algebras are closely related (see e.g., the table on page 29 there), there is a good chance that one might be able to view the Klein subgroup or a suitable extension thereof as the Clifford group of an appropriate algebra, and thereby obtain a connection between their representations.