We study the defect texture in symmetric and asymmetric states of a nematic-liquid-crystal shell with conic and planar degenerate surface anchorings on the inner and outer spherical boundaries, respectively. To achieve the equilibrium nematic orientation, we numerically minimize the Landau-de Gennes free energy by employing surface potentials on the shell walls. The symmetric nematic shells energetically have stable configurations independent of thickness. In thick shells, the director field satisfies bipolar and hexadecapolar configurations between boundaries. In thin shells, the boojums transform into two stable disclination curves.