We propose a new framework for homonuclear dipolar decoupling in solid-state NMR that provides a theoretical link between the FSLG, PMLG and DUMBO families. We show that through the use of a Legendre polynomial basis, the phase modulation of these decoupling schemes can be described by the same set of parameters, permitting for the first time a direct theoretical comparison between these methods. Use of this common basis reveals that the central decoupling mechanism is the same for DUMBO and FSLG/PMLG and that a similar vector picture can be used to describe both methods. In addition to the common root of decoupling efficiency, this new analysis highlights two major points of difference between the methods. First, the DUMBO phase modulation consists not only of a linear change in phase with time a la PMLG but also smaller high-order oscillations, which act to improve line-narrowing performance. Second, we show how the DUMBO phase waveforms are generated from a four-step permutation of a single asymmetric unit, in contrast to the two-step permutation of PMLG. Numerical simulations and experimental results suggest that this latter point of difference is responsible for the superior performance of DUMBO in the presence of significant RF inhomogeneity.