The complex spin order of hyperpolarized multispin systems giving rise to anomalous NMR spectral patterns that vary with the RF excitation angle is analyzed by decomposing its nutation behavior in a superposition of Fourier harmonics. The product operator formalism is applied to calculating the spectral contributions of the various mutual alignments of scalar coupled spins. Two cases are treated, namely systems exhibiting only differences in population of their spin states and systems showing in addition zero-quantum coherences between states, a situation often seen at hyperpolarization. After deriving the general solution a number of representative examples are discussed in detail. The theoretical treatment is applied to analyzing the spin order observed in a hyperpolarized two-spin system that is prepared in the singlet state by para-hydrogen induced polarization.