We calculate the proton and neutron spin contributions for nuclei using semi-empirical methods, as well as a novel hybrid ab initio/semi-empirical method, for interpretation of experimental data. We demonstrate that corepolarisation corrections to ab initio nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. We derive constraints on the spin-dependent P,T-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We derive a limit on the CPT and Lorentzinvariance-violating parameter |b p ⊥ | < 7.6 × 10 −33 GeV, which improves on the most stringent existing limit by a factor of 8, and we demonstrate sensitivities to the parametersd p ⊥ andg p D⊥ at the level ∼10 −29 -10 −28 GeV, which is a one order of magnitude improvement compared to the corresponding existing sensitivities. We extend previous analysis of nuclear anapole moment data for Cs to obtain new limits on several other CPT and Lorentz-invariance-violating parameters: |b p 0 | < 7×10 −8 GeV, |d p 00 | < 8×10 −8 , |b n 0 | < 3×10 −7 GeV and |d n 00 | < 3 × 10 −7 .