The interaction of atoms, molecules, crystals and nanotubes as well as of artificial nanostructures with time-periodic laser fields can lead to high-order space-time symmetries. These space-time symmetries (also referred to as dynamical symmetries (DSs)) are to the Floquet Hamiltonian what spatial symmetries are to the field-free Hamiltonian. Consequently, DS properties of Floquet states can be studied by group theory. Here we use DSs to characterize the modifications of electronic energy levels induced by circularly polarized laser fields in quantum rings, thin crystals and carbon nanotubes (CNTs). Furthermore, we show that DSs underlie the formulation of selection rules for high-order harmonic generation processes in strong laser fields. This enables one to predict outputs of harmonic generation experiments and even suggests new experiments which would lead to 'engineered' spectra. By doing so, we place DS analysis of harmonic generation processes in high-intensity fields on a firm foundation, analogously to that possessed by symmetry analysis in 'conventional' spectroscopy. This formalism is applied to single-walled CNTs interacting with circularly polarized laser fields. We find that CNTs can be excellent candidates for a selective and efficient generation of high-order harmonics, up to the soft X-ray regime.