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We obtain a new relation between the distributions$$\upmu _t$$${\mu }_t$at different times$$t\ge 0$$$t\ge 0$of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions$$\upmu _t$$${\mu }_t$backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving$$\upmu _t$$${\mu }_t$which in turn brings new identities for expectations with respect to$$\upmu _t$$${\mu }_t$. The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.