We study the influence of the quadratic Zeeman effect on the Mott-insulator phases of hard-core 1D spin-$3/2$ fermions. We show that, contrary to spinor bosons, the quadratic Zeeman coupling preserves an $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ symmetry, leading for large-enough quadratic Zeeman coupling to an isotropic pseudo-spin-$1/2$ Heisenberg antiferromagnet. Decreasing the quadratic Zeeman coupling, this phase undergoes, depending on the scattering lengths, either a Kosterlitz-Thouless transition into a gapped dimerized phase or a commensurate-incommensurate transition into a gapless spin liquid. This rich phase diagram can be observed experimentally in four-component fermions in optical lattices under similar entropy constraints to those needed for N\'eel order in spin-$1/2$ gases.