Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be a product space ${\mathbb{R}}^{\mathbb{N}}$ equipped with the product $\sigma$-algebra $\mathcal{B} ({\mathbb{R}}^{\mathbb{N}})$. Let $(X_n)_{n \geq 1}$ be the coordinate process on $\Omega$, i.e. \begin{equation} \omega = (X_1 (\omega), X_2 (\omega), \ldots). \end{equation} Moreover, let $(X_n)_{n \geq 1}$ be exchangeable, i.e. the process $(X_ {\tau(n)})_{n \geq 1}$ has the same law as $(X_ {n})_{n \geq 1}$ under $\mathbb{P}$ for any permutation $\tau$ of $\mathbb{N}$ which fixes all but finitely many elements of $\mathbb{N}$.
Let $S$ be the $\sigma$-algebra on $\Omega$ made up of events $A \in \mathcal{B} ({\mathbb{R}}^{\mathbb{N}})$ such that \begin{equation} (X_1 (\omega), X_2 (\omega), \ldots ) \in A \quad \text{iff} \quad (X_{\tau (1)} (\omega), X_{\tau(2)} (\omega), \ldots ) \in A. \end{equation}
Show that for any $A \in S$, \begin{equation} \mathbb{E} ( {\mathbf{1}}_A | \, \sigma ( X_1, X_2, \ldots, X_n ) ) = \mathbb{E} ( {\mathbf{1}}_A | \, \sigma ( X_{n+1}, X_{n+2}, \ldots, X_{2n} ) ). \end{equation} (I only know that we need to use the permutation $\tau$ that interchanges $1$ and $n+1$, $2$ and $n+2$, etc, but I don't know how to prove this fact rigorously.)
A starting point could be the following: if $\tau$ is a permutation with finite support, then for each bounded random variable $Y$ and each $\sigma$-algebra $\mathcal M$, we have $$\mathbb E[Y\mid\mathcal M](\tau(\omega))=\mathbb E[Y^\tau\mid \tau^{-1}\mathcal M](\omega),$$ where $Y^\tau$ denotes the random variable defined by $Y^\tau:=Y(\tau(\omega))$.