We establish bijections between three classes of combinatorial objects that have been studied in very different contexts: lattice walks in simplicial regions as introduced by Mortimer--Prellberg, standard cylindric tableaux as introduced by Gessel--Krattenthaler and Postnikov, and sequences of states in the totally asymmetric simple exclusion process. This perspective allows us to translate symmetries from one setting into another, revealing unexpected properties of these objects. Specifically, we show that a recent bijection of Courtiel, Elvey Price and Marcovici between certain simplicial walks with forward and backward steps is equivalent to a cylindric analogue of the Robinson--Schensted correspondence. Originally defined by Neyman by iterating an insertion operation, we provide an alternative description of this correspondence by introducing a cylindric version of Fomin's growth diagrams. This natural description elucidates the symmetry of the correspondence, and it allows us to interpret the above walks as oscillating cylindric tableaux.