In solid-state NMR, many powerful pulse sequences under the condition of magic-angle spinning can be analyzed on the basis of the C- and R-sequences developed by Levitt and co-workers. It has been speculated for some years that the basic elements commonly used in symmetry-based recoupling pulse sequences have certain kind of internal symmetries. We show by a detailed analysis that a set of internal selection rules does exist for many basic elements. These internal selection rules may allow a more versatile design of CN(n)(ν) or RN(n)(ν) sequences when n is an integer or half-integer multiple of N. As an illustration, we have derived the symmetry arguments to rationalize the observation that the C-REDOR pulse sequence can suppress homonuclear dipole-dipole interaction, leading to the design of new windowed basic elements usable for heteronuclear dipolar recoupling with active suppression of homonuclear dipole-dipole interaction. Numerical simulations and experiments measured for [U-(13)C,(15)N]-L-alanine have been used to validate our approach. On a more general note, the symmetry rules discussed in this work can also be applied for the design of supercycles.