The author considers processes on a countable state space, indexed by the integers, which are time homogeneous Markov processes in both directions of time. Such a process can be constructed from given forward and backward (substochastic) transition matrices p and q, together with a family of one-dimensional distributions \(\{\pi_ n\}\), satisfying \(\pi_ n(i)p(i,j)=\pi_{n+1}(j)q(j,i).\) Under a kind of irreducibility requirement on p, the author describes a type of periodic dependence among the \(\{\pi_ n\}\). The special case which occurs when \(\pi_ 0p\leq \rho \pi_ 0\) for some \(\rho >0\), termed ”quasistationarity”, is studied in detail and compared to the ”extended chains” of J. G. Kemeny, J. L. Snell and A. W. Knapp, Denumerable Markov chains. 2nd ed. (1976; Zbl 0348.60090).