  Let $G_1$ and $G_2$ be discrete subgroups of $SO(3)$. The double quotients of
the form $X(G_1,G_2)=G_1\backslash SO(3)/G_2$ were introduced in material
science under the name misorientation spaces. In this paper we review several
known results that allow to study topology of misorientation spaces. Neglecting
the orbifold structure, all misorientation spaces are closed orientable
topological 3-manifolds with finite fundamental groups. In case when $G_1,G_2$
are crystallography groups, we compute the fundamental groups
$\pi_1(X(G_1,G_2))$, and apply Thurston's elliptization conjecture to describe
these spaces. Many misorientation spaces are homeomorphic to $S^3$ by
Poincar\'{e} conjecture. The sphericity in these examples is related to the
theorem of Mikhailova--Lange, which constitutes a certain real analogue of
Chevalley--Shephard--Todd theorem. We explicitly describe topological types of
several misorientation spaces avoiding the reference to Poincar\'{e}
conjecture. Classification of misorientation spaces allows to introduce new
$n$-valued group structures on $S^3$ and $\mathbb{R}P^3$. Finally, we outline
the connection of the particular misorientation space $X(D_2,D_2)$ to
integrable dynamical systems and toric topology.
