The average Hamiltonian theory (AHT) of several classes of symmetry-based radio-frequency pulse sequences is developed to second order, allowing quantitative analyses of a wide range of recoupling and decoupling applications in magic-angle-spinning solid state nuclear magnetic resonance. General closed analytical expressions are presented for a cross term between any two interactions recoupled to second order AHT. We classify them into different categories and show that some properties of the recoupling pulse sequence may be predicted directly from this classification. These results are applied to examine a novel homonuclear recoupling strategy, effecting a second order average dipolar Hamiltonian comprising trilinear triple quantum (3Q) spin operators. We discuss general features and design principles of such 3Q recoupling sequences and demonstrate by numerical simulations and experiments that they provide more efficient excitation of C-13 3Q coherences compared to previous techniques. We passed up to 15% of the signal through a state of 3Q coherence in rotating powders of uniformly C-13-labeled alanine and tyrosine. Second order recoupling-based C-13 homonuclear 3Q correlation spectroscopy is introduced and demonstrated on tyrosine. (C) 2004 American Institute of Physics.