Let (M-t : t > 0) be a Markov process of tessellations of R-e, and let (C-t : t > 0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (a(t )Ca(t ): t is an element of R), a > 1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.