On the $\sigma$-algebra of symmetrical events for a countable Markov chain

It is proved that the $\sigma$-algebra of symmetrical events for a countable Markov chain $\{x_n, n\ge 1\}$ coincides a. s. with $T(\sigma_1\bigcap\sigma_2)$, where $T$ is a left shift and $\sigma_i=\sigma(x_i)$, $i=1,2$.