We investigate the properties of conduction electrons in single-walled armchair carbon nanotubes (SWNT) in the presence of both transverse electric and magnetic fields. We find that these fields provide a controlled means of tuning low-energy band structure properties such as inducing gaps in the spectrum, breaking various symmetries and altering the Fermi velocities. We show that the fields can strongly affect electron-electron interaction, yielding tunable Luttinger liquid physics, the possibility of spin-charge-band separation, and a competition between spin-density-wave and charge-density-wave order. For short tubes, the fields can alter boundary conditions and associated single-particle level spacings as well as quantum dot behavior.