  We establish a symmetry in a variety of integrable stochastic systems:
Certain multi-point distributions of natural observables are unchanged under a
shift of a subset of observation points. The property holds for stochastic
vertex models, (1+1)d directed polymers in random media, last passage
percolation, the Kardar-Parisi-Zhang equation, and the Airy sheet. In each
instance it leads to computations of previously inaccessible joint
distributions. The proofs rely on a combination of the Yang-Baxter
integrability of the inhomogeneous colored stochastic six-vertex model and
Lagrange interpolation. We also show that a simplified (Gaussian) version of
our theorems is related to the invariance in law of the local time of the
Brownian bridge under the shift of the observation level.
