We introduce O-systems (Definition 3.1) of orthogonal transformations of ℝ m , and establish ℝ correspondences both between equivalence classes of Clifford systems and those of O-systems, and between O-systems and orthogonal multiplications of the form μ : ℝ n × ℝ m → ℝ m , which allow us to solve the existence problems both for O -systems and for umbilical quadratic harmonic morphisms simultaneously. The existence problem for general quadratic harmonic morphisms is then solved by the Splitting Lemma. We also study properties possessed by all quadratic harmonic morphisms for fixed pairs of domain and range spaces.