  We obtain a new relation between the distributions $\mu_t$ at different times
$t\ge 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion
Process) 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 $\mu_t$ 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
$\mu_t$ which in turn brings new identities for expectations with respect to
$\mu_t$.
  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.
