Promotion permutations have recently been associated to each rectangular standard Young tableau by Gaetz--Pechenik--Pfannerer--Striker--Swanson. Here we relate promotion permutations to the Robinson--Schensted (RS) correspondence. More precisely, we show that taking a pair of standard Young tableaux of the same rectangular shape, stacking them, and computing the middle promotion permutation yields the RS permutation of the pair up to simple twists. Moreover, the full list of promotion permutations in this special case encodes Viennot's geometric shadow line construction. As a consequence, we characterize a subset of the collection of possible promotion permutations in terms of crossing and nesting numbers.