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The $\bf{g}$ tensor, which determines the reaction of Kramers-degenerate states to an applied magnetic field, is of increasing importance in the current design of spin qubits. It is affected by details of heterostructure composition, disorder, and electric fields, but it inherits much of its structure from the effect of the spin-orbit interaction working at the crystal-lattice level. Here we uncover new symmetry and topological features of $\bf{g}=\bf{g}_L+\bf{g}_S$ for important valence and conduction bands in silicon, germanium, and gallium arsenide. For all crystals with high (cubic) symmetry, we show that large departures from the nonrelativistic value $g=2$ are guaranteed by symmetry. In particular, considering the spin part $\bf{g}_S(\bf{k})$, we prove that the scalar function $det(\bf{g}_S(\bf{k}))$ must go to zero on closed surfaces in the Brillouin zone, no matter how weak the spin-orbit coupling is. We also prove that for wave vectors $\bf{k}$ on these surfaces, the Bloch states $|u_{n\bf{k}}\rangle$ have maximal spin-orbital entanglement. Using tight-binding calculations, we observe that the surfaces $det(\bf{g}(\bf{k}))=0$ exhibit many interesting topological features, exhibiting Lifshitz critical points as understood in Fermi-surface theory., Comment: 15 pages, 8 figures