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 $\mu :{ℝ}^n\times {ℝ}^m\to {ℝ}^m$, 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...