Using Henriques' and Kamnitzer's cactus groups, Schützenberger's promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals. For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram's map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general linear groups. We prove these results by applying van Leeuwen's generalisation of Fomin's local rules for jeu de taquin, connected to the action of the cactus groups by Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted correspondence. This work is based on a joint research project with Martin Rubey and Bruce W. Westbury. In chapter 1 we give a general introduction and state related work. Chapter 2 connects the algebraic world of representations with combinatorics and we present our findings in chapter 3. In chapter 4 we define promotion and evacuation as actions of certain elements of a cactus group and state local rules for algorithmically calculating these actions. The local rules are strongly related to the rules of our growth diagram bijections from chapter 5. The last chapter 6 is meant for proofs only. Chapters 1, 3, 4, 5 and 6 are also published separately as a joint paper.