We explore the effects of symmetry on the performance of phase-modulated homonuclear dipolar decoupling in H-1 solid-state NMR. We demonstrate that the symmetry of the DUMBO family of decoupling sequences is the result of two well-defined symmetry expansions. The first is an antipalindromic expansion that arises from the symmetrization step that was built into the original architecture of the DUMBO sequence. The second is a mirror-pair expansion that inverts the sign of the phase modulation in the second half of the pulse sequence relative to the first. The combination of these two symmetry expansions generates a sequence of four Lee-Goldburg-type rotations in the rotating frame. The axes of rotation, oriented at the magic angle, are separated in the transverse plane by 2 alpha, where alpha is chosen to minimize the sensitivity of the sequence to instrument imperfections such as rf inhomogeneity. The efficiency of the DUMBO symmetry for decoupling is demonstrated experimentally, and the effect of the alpha-phase-shift parameter is investigated. A new decoupling sequence (LG4) that combines the DUMBO symmetry with alpha = 55 degrees is introduced and is shown to produce very efficient decoupling as well as a nearly 2-fold increase in coherence lifetimes when compared to standard PMLG/FSLG decoupling.