We study the duality relationship between M-theory and heterotic string theory at the classical level, emphasising the transformations between the Kaluza-Klein reductions of these two theories on the K3 and T3 manifolds. Particular attention is devoted to the corresponding structures of σ-model cosets and the correspondence between the p-brane charge lattices. We also present simple and detailed derivations of the global symmetries and coset structures of the toroidally compactified heterotic theory in all dimensions D ⩾ 3, making use of the formalism of solvable Lie algebras.