Deformations of single-walled carbon nanotubes (SWNTs) change their band structure in the nanoelectromechanical systems. In this study, we investigated the response of the electronic structure of chiral and nonchiral SWNTs (8,7), (9,6), (10,5), (7,7), (11,0), (12,0), and (13,0) to twisting and axial tension modes by using the symmetrized linear augmented cylindrical wave technique. We showed that perturbations of the band structures depend on a “family” index p = (n1 − n2)mod 3 (where p = −1, 0 or 1). Twisting the semiconducting (8,7) tubule with p = 1 in the direction of the screw axis is accompanied by the large broadening of minimum gap E11 and narrowing of the second gap E22, while these gaps drastically change in the tubule (10,5) with p = −1. In these tubules, changing the direction of twisting leads to the reversal in direction of the gap shifts. Regardless of the twisting direction, in metallic (7,7) and quasi-metallic (9,6) SWNTs with p = 0, the E11 gap rapidly increases from 0.0 and 0.035 eV to about 1 eV. When twisting the zigzag tubules (13,0) p = 1 and (11,0) p = −1, the gaps E11 equal to about 0.8 eV increase and decrease by several hundredths of eV. On the contrary, the compression and extension of these tubules cause a sharp change in their band structure with approximately a twofold change in the gaps E11 and E22 and inversion in the sequence of the boundary bands. The similar deformation of the armchair nanotube (7,7) has practically no effect on its electronic levels. In the case of zigzag (12,0) p = 0 SWNT, all deformation modes transform the quasi-metallic tubule into the semiconductor.