In this thesis we use homotopy-theoretic techniques to establish a range of combinatorially-governed relations in the algebraic invariants of polyhedral product spaces.First, for a flag simplicial complex K, we specify a necessary and sufficient combinatorial condition for the commutator subgroup RCK of a right-angled Coxeter group, which is the fundamental group of the real moment-angle complex RK, to be a one-relator group; and for the loop homology algebra H*(Ω ZK) of the moment-angle complex ZK to be a one-relator algebra. This moreover establishes a combinatorial link between distinct concepts of geometric group theory and homotopy theory.Second, we give a substantial generalisation of the Whitehead product to a construction called the higher Whitehead map, which takes maps from homotopy sets of the form [Σ X,Y] to a new map in homotopy sets related to polyhedral products. We analyse these maps systematically via the combinatorial structure underlying the polyhedral products involved, and derive combinatorial conditions describing when these maps are non-trivial. Moreover, we establish non-trivial relations between higher Whitehead maps which are governed combinatorially. These relations greatly generalise the Jacobi identity for Whitehead products, and results of Hardie on relations among exterior Whitehead products.