Hello,
in case you work with DFT or SCF, you can get the total wave-function symmetry from the multiplication table of irreps of the respective point group. Closed shells (by shell, I mean all MOs of the same energy, of course) should give only the totally symmetric repre, so doing this manually might be feasible. But I would hope for some script which would do that. In case of the excited states, Turbomole inform about the symmetry, but for the ground state, I also had problem to find it in output.
In case, the wave-function consists of more Slater determinants, one may chose one determinant and in case more possible symmetries will result from him (product of two irreps might be reducible), one might look whether there is or is not linear combination of determinants with only one symmetry (it should be).
Best regards,
Jakub