  We study the twist-3 spin-orbit correlations of quarks described by the
nucleon matrix elements of the parity-odd rank-2 tensor QCD operator (the
parity-odd partner of the QCD energy-momentum tensor). Our treatment is based
on the effective dynamics emerging from the spontaneous breaking of chiral
symmetry and the mean-field picture of the nucleon in the large-$N_c$ limit.
The twist-3 QCD operators are converted to effective operators, in which the
QCD interactions are replaced by spin-flavor-dependent chiral interactions of
the quarks with the pion field. We compute the nucleon matrix elements of the
twist-3 effective operators and discuss the role of the chiral interactions in
the spin-orbit correlations. We derive the first-quantized representation in
the mean-field picture and develop a quantum-mechanical interpretation. The
chiral interactions give rise to new spin-orbit couplings and qualitatively
change the correlations compared to the quark model picture. We also derive the
twist-3 matrix elements in the topological soliton picture where the quarks are
integrated out (skyrmion). The methods used here can be extended to other QCD
operators describing higher-twist nucleon structure and generalized parton
distributions.
