We extend the recent radial symmetry results by Pisante [J. Funct. Anal., 260 (2011), pp. 892--905] and Millot and Pisante [J. Eur. Math. Soc. $($JEMS$)$, 12 (2010), pp. 1069--1096] (who show that the equivariant solutions are the only entire solutions of the three-dimensional Ginzburg--Landau equations in superconductivity theory) to the Landau--de Gennes framework in the theory of nematic liquid crystals. In the low temperature limit, we obtain a characterization of global Landau--de Gennes minimizers, in the restricted class of uniaxial tensors, in terms of the well-known radial-hedgehog solution. We use this characterization to prove that global Landau--de Gennes minimizers cannot be purely uniaxial for sufficiently low temperatures.