  A $D_{\infty}$-topological Markov chain is a topological Markov chain
provided with an action of the infinite dihedral group $D_{\infty}$. It is
defined by two zero-one square matrices $A$ and $J$ satisfying
$AJ=JA^{\textsf{T}}$ and $J^2=I$. Flip signature is obtained from symmetric
bilinear forms with respect to $J$ on the eventual kernel of $A$. We modify
Williams' decomposition theorem to prove flip signature is a
$D_{\infty}$-conjugacy invariant. We introduce natural $D_{\infty}$-actions on
Ashley's eight-by-eight and the full two-shift. The Flip signatures show that
Ashley's eight-by-eight and the full two-shift equipped with the natural
$D_{\infty}$-actions are not $D_{\infty}$-conjugate. We also discuss the notion
of $D_{\infty}$-shift equivalence and the Lind zeta function.
