  We calculate the proton and neutron spin contributions for nuclei using
semi-empirical methods, as well as a novel hybrid \emph{ab
initio}/semi-empirical method, for interpretation of experimental data. We
demonstrate that core-polarisation corrections to \emph{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 Lorentz-invariance-violating parameter
$|\tilde{b}_{\perp}^p| < 7.6 \times 10^{-33}$ GeV, which improves on the most
stringent existing limit by a factor of 8, and demonstrate sensitivities to the
parameters $\tilde{d}_{\perp}^p$ and $\tilde{g}_{ D\perp}^p$ at the level $\sim
10^{-29} - 10^{-28}$ 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: $\left|b_0^p \right| < 7
\times 10^{-8}$ GeV, $\left|d_{00}^p \right| < 8 \times 10^{-8}$, $\left|b_0^n
\right| < 3 \times 10^{-7}$ GeV and $\left|d_{00}^n \right| < 3 \times
10^{-7}$.
