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Structural and magnetic symmetry breakings could lead to insulating phases in open d-shell transition metal oxides that could otherwise stay metallic. Here, we find from density functional theory supercell (as well as molecular dynamics) calculations how structural symmetry breaking of V-V dimerization and magnetic spin symmetry breaking shape the insulator-to-metal transition in VO₂. The V-V dimerization tends to remove the magnetic moments of the dimerized V⁴⁺ ions whose occupied d states form a spin-singlet, and the magnetism tends to weaken the V-V dimerization. In the high-temperature metallic rutile phase of VO₂, the V-V dimers are uncoupled leading to unavoidable local magnetic moments on the V⁴⁺ ions with residual spin splitting, according to DFT supercell calculations. This type of competitive interaction between the structural and magnetic symmetry breakings could deepen the understanding of the insulator-to-metal transition mechanism in open d-shell transition metal oxides and be utilized to realize a type of structural-magnetic logic devices.