Some new symmetry theorems are presented which simplify the task of designing multiple-pulse radio-frequency pulse sequences in magic-angle-spinning solid-state NMR. The symmetry theorems apply to sequences denoted RNnv which consists of N repetitions of a pulse sequence element R, alternating in phase between the values +/- pi v/N. Each R element ideally rotates the spins by an angle pi about the rotating frame x-axis. The entire RNnv sequence is timed to span n rotational periods. Applications are presented for homonuclear double-quantum and zero-quantum recoupling, heteronuclear decoupling and heteronuclear recoupling.