Many integrable stochastic particle systems in one space dimension (like TASEP) remain integrable when we equip each particle with its own "speed" parameter. Moreover, the distribution of the particle system displays a certain symmetry in these parameters. I will discuss probabilistic consequences of this symmetry in a number of models including q-TASEP and random polymers. In particular, we discover a new continuous time dynamics preserving the distribution of the q-TASEP (with any fixed time parameter, and started from the step initial configuration). We also describe the dual process to the stationary dynamics which is a certain transient modification of the stochastic q-Boson system.
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