  We introduce a notion of $\lambda$-graph bisystem. It consists of a pair
$({\frak L}^-, {\frak L}^+)$ of two labeled Bratteli diagrams ${\frak L}^-,
{\frak L}^+$ over alphabets $\Sigma^-, \Sigma^+$, respectively, and satisfy
certain compatibility condition of their labeling on edges. Its matrix
presentation is called a symbolic matrix bisystem. We first show that any
$\lambda$-graph bisystem presents subshifts and conversely any subshift is
presented by a $\lambda$-graph bisystem, called the canonical $\lambda$-graph
bisystem for the subshift. We introduce a notion of properly strong shift
equivalence on symbolic matrix bisystems and show that two subshifts are
topologically conjugate if and only if their canonical symbolic matrix
bisystems are properly strong shift equivalent. A $\lambda$-graph bisystem
$({\frak L}^-, {\frak L}^+)$ yields a pair of $C^*$-algebra written
${{\mathcal{O}}_{{\frak L}^-}^+}, {{\mathcal{O}}_{{\frak L}^+}^-}$. We show
that the $C^*$-algebras are universal unique $C^*$-algebras subject to certain
operator relations among canonical generators encoded by $\lambda$-graph
bisystem $({\frak L}^-, {\frak L}^+).$ If a $\lambda$-graph bisystem comes from
a $\lambda$-graph system of a finite directed graph, then the associated
subshift is the two-sided topological Markov shift $(\Lambda_A, \sigma_A)$ by
its transition matrix $A$ of the graph, and the associated $C^*$-algebra
${{\mathcal{O}}_{{\frak L}^-}^+}$ is isomorphic to ${\mathcal{O}}_A,$ whereas
the other $C^*$-algebra ${{\mathcal{O}}_{{\frak L}^+}^-}$ is isomorphic to
$C(\Lambda_A)\rtimes_{\sigma_A^*}\mathbb{Z}$ of the commutative $C^*$-algebra
$C(\Lambda_A)$ on $\Lambda_A$ by the automorphism induced by the homeomorphism
of the shift $\sigma_A.$ This phenomena shows a duality between
${\mathcal{O}}_A$ and $C(\Lambda_A)\rtimes_{\sigma_A^*}\mathbb{Z}$.
