  We present a detailed theoretical analysis of the Wigner crystal states in
confined semiconducting carbon nanotubes. We show by robust scaling arguments
as well as by detailed semi-microscopic calculations that the effective
exchange interaction has an SU(4) symmetry, and can reach values even as large
as $J\sim 100 {\rm \,K}$ in weakly screened, small diameter nanotubes, close to
the Wigner crystal - electron liquid crossover. Modeling the nanotube carefully
and analyzing the magnetic structure of the inhomogeneous electron crystal, we
recover the experimentally observed 'phase boundaries' of Deshpande and
Bockrath [V. V. Deshpande and M. Bockrath, Nature Physics $\mathbf 4$, 314
(2008)]. Spin-orbit coupling only slightly modifies these phase boundaries, but
breaks the spin symmetry down to SU(2)$\times$SU(2), and in Wigner molecules it
gives rise to interesting excitation spectra, reflecting the underlying SU(4)
as well as the residual SU(2)$\times$SU(2) symmetries.
