  For a homeomorphism $T \colon X \to X$ of a Cantor set $X$, the mapping class
group $\mathcal{M}(T)$ is the group of isotopy classes of
orientation-preserving self-homeomorphisms of the suspension $\Sigma_{T}X$. The
group $\mathcal{M}(T)$ can be interpreted as the symmetry group of the system
$(X,T)$ with respect to the flow equivalence relation. We study
$\mathcal{M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We
show that when $(X,T)$ is a subshift associated to a substitution, the group
$\mathcal{M}(T)$ is an extension of $\mathbb{Z}$ by a finite group; for a large
class of substitutions including Pisot type, this finite group is a quotient of
the automorphism group of $(X,T)$. When $(X,T)$ is a minimal subshift of linear
complexity satisfying a no-infinitesimals condition, we show that
$\mathcal{M}(T)$ is virtually abelian. We also show that when $(X,T)$ is
minimal, $\mathcal{M}(T)$ embeds into the Picard group of the crossed product
algebra $C(X) \rtimes_{T} \mathbb{Z}$.
