  We study the periodic $q$-Whittaker and Hall-Littlewood processes, two
probability measures on sequences of partitions. We prove that a certain
observable of the periodic $q$-Whittaker process exhibits a $(q,u)$ symmetry
after a random shift, generalizing a previous result of Imamura, Mucciconi, and
Sasamoto who showed a matching between the periodic Schur and $q$-Whittaker
measures, and also give a vertex model formulation of their result. As part of
our proof of the $(q,u)$ symmetry, we obtain contour integral formulas for both
the periodic $q$-Whittaker and Hall-Littlewood processes. We also show a
matching between certain observables in the periodic Hall-Littlewood process
and in a quasi-periodic stochastic six vertex model after a suitable random
shift, and discuss a limit to the stationary periodic stochastic six vertex
model.
