  We discover a new property of the stochastic colored six-vertex model called
flip-invariance. We use it to show that for a given collection of observables
of the model, any transformation that preserves the distribution of each
individual observable also preserves their joint distribution. This generalizes
recent shift-invariance results of Borodin-Gorin-Wheeler. As limiting cases, we
obtain similar statements for the Brownian last passage percolation, the
Kardar-Parisi-Zhang equation, the Airy sheet, and directed polymers. Our proof
relies on an equivalence between the stochastic colored six-vertex model and
the Yang-Baxter basis of the Hecke algebra. We conclude by discussing the
relationship of the model with Kazhdan-Lusztig polynomials and positroid
varieties in the Grassmannian.
