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 RNnν, where N, n and ν are integral symmetry numbers. A RNnν sequence consists of N repetitions of a pulse sequence element R, alternating in phase between the values ±πν/N. Each R element ideally rotates the spins by an angle π about the rotating frame x-axis. The entire RNnν sequence is timed to span n rotational periods. The symmetry theorems allow the construction of sequences suitable for a wide range of purposes in solid-state NMR. Specific applications are presented for homonuclear double-quantum recoupling, heteronuclear decoupling and heteronuclear recoupling. For the case of homonuclear double-quantum recoupling, we demonstrate a high-performance sequence with the symmetry R1426. For the case of heteronuclear decoupling, we show that the existing theory provides a framework for understanding the operation of the two-pulse phase modulation (TPPM) scheme (A. E. Bennett, C. M. Rienstra, M. Auger, K. V. Lakshmi and R. G. Griffin, J. Chem. Phys. 103, 6951 (1998)). We also demonstrate a heteronuclear dipolar recoupling sequence with the symmetry R1817.