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 $Jsim 100 {rm ,K}$ 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.
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