The iron-pnictide superconductors have generated tremendous excitement as the competition between magnetism and superconductivity has allowed unique in-roads towards elucidating a microscopic theory of unconventional high-temperature superconductivity. In addition to the stripe spin density wave (C-2M(a)) phase observed in the parent compounds of all iron-pnictide superconductors, two novel magnetic orders have recently been discovered in different parent structures: an out-of-plane collinear double-Q (C-4M(c)) structure in the holedoped (Ca, Sr, Ba)(1-x)(Na)(x)Fe2As2 and Ba1-xKxFe2As2 families, and a spin vortex crystal "hedgehog" (C-4M(ab)) structure in the CaKFe4As4 family. Using neutron diffraction, we demonstrate that LaFeAs1-xPxO contains all three magnetic orders within a single-phase diagram as a function of substitution, all of which compete strongly with superconductivity. Our experimental observations combined with theoretical modeling demonstrate how the reduction in electronic correlations by chemical substitution results in larger Fermi surfaces and the sequential stabilization of multiple magnetic anisotropies. Our work presents a unified narrative for the competing magnetic and superconducting phases observed in various iron-pnictide systems with different crystal structures and chemistry.