quantum chemistry - What constraints are imposed on a wavefunction by the symmetry of the system?

As a follow-up to my answer here, I'd like to ask what exactly does it mean for a wavefunction to "respect the symmetry" of the system.

The original context is: immediately after ionisation of $\ce{CH4}$, the wavefunction of $\ce{CH4+}(T_\mathrm{d})$ cannot be adequately described by removal of an electron from an $\mathrm{sp^3}$-type hybrid orbital, as the resulting Slater determinant $|\phi_1\phi_2\phi_3\phi_4\overline{\phi}_1\overline{\phi}_2\overline{\phi}_3|$ does not reflect the tetrahedral symmetry of the system immediately after ionisation ($\phi_i$ being the four hybrid orbitals).

Intuitively, it is clear that this is related to the fact that the four C–H bonds must be equivalent. Mathematically, how is this formalised, i.e. what is the criterion that the wavefunction must obey?

As a side question, how does the criterion above apply to a boron atom, for example? Is the wavefunction $\mathrm{1s}^2 \mathrm{2s}^2 \mathrm{2p}_x^1$ acceptable on symmetry grounds? My gut feeling is that it's not (because the electron density needs to be spherically symmetric?), but I might really be overthinking this.