I'm trying to learn Teichmüller theory, but appear to get stuck early on. Let $\Sigma$ be a smooth closed oriented surface of genus $g\geqslant 2$ and let $\mathrm{Conf}(\Sigma)$ denote the set of smooth conformal structures on $\Sigma$. The choice of a complex structure $J$ turns $\Sigma$ into a hyperbolic Riemann surface $X$ and identifies $\mathrm{Conf}(\Sigma)$ with the set $$ \mathcal{B}(X)=\left\{\mu \in \Omega^{0,1}(X,K_X^{-1}) : |\mu|<1\right\} $$ of Beltrami differentials on $X$. The group $\mathrm{Diff}_0(\Sigma)$ of diffeomorphisms isotopic to the identity acts on $\mathrm{Conf}(\Sigma)$ by pullback and hence we get an induced action of $\mathrm{Diff}_0(\Sigma)$ on $\mathcal{B}(X)$ making the bijection $\mathcal{B}(X)\simeq \mathrm{Conf}(\Sigma)$ equivariant.
Teichmüller space is the quotient $\mathcal{T}_g=\mathrm{Conf}(\Sigma)/\mathrm{Diff}_0(\Sigma)$. Given the parametrisation of Teichmüller space (by harmonic maps/Higgs bundles techniques) in terms of holomorphic quadratic differentials on $X$, I'm tempted to think that every orbit of the action of $\mathrm{Diff}_0(\Sigma)$ on $\mathcal{B}(X)$ has a unique representative consisting of a harmonic Beltrami differential, that is, an element of $$ \mathcal{H}(X)=\left\{\mu \in \mathcal{B}(X) : \mu=\overline{q}/\sigma,\; q \in H^0(X,K_X^2) \right\}, $$ where $\sigma$ denotes the hyperbolic metric of $J$. Is this correct?