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This paper introduces the double travelling salesman problem with multiple stacks and presents four different metaheuristic approaches to its solution. The double TSP with multiple stacks is concerned with determining the shortest route performing pickups and deliveries in two separated networks (one for pickups and one for deliveries) using only one container. Repacking is not allowed, instead each item can be positioned in one of several rows in the container, such that each row can be considered a LIFO (last in, first out) stack, but no mutual constraints exist between the rows. Two different neighbourhood structures are developed for the problem and used with each of three local search metaheuristics. Additionally some simpler removal and reinsertion operators are used in a Large neighbourhood search framework. Finally some computational results are given along with lower bounds on the objective value.