The case of C-13-labeled chloroform exchanging between a free site in solution and the encaged site within the cryptophane-D cavity is investigated using the measurements of longitudinal cross-correlated relaxation rates, involving the interference of the dipole-dipole and chemical shielding anisotropy interactions. A compact theoretical expression is provided, along with an experimental protocol, based on INEPT (insensitive nuclei enhanced by polarization)enhanced double-quantum-filtered inversion recovery measurements. The analysis of the build-up curves results in a set of cross-correlated relaxation rates for both the C-13 and H-1 spins at the two sites. It is demonstrated that the results can be given a consistent interpretation in terms of molecular-level properties, such as rotational correlation times, the Lipari-Szabo order parameter, and interaction strength constants. The analysis yields the bound-site carbon-13 chemical shielding anisotropy, Delta sigma(C) = -58 +/- 8 ppm, in good agreement with most recent liquid-crystal measurements and the corresponding proton shielding anisotropy, Delta sigma(H) = 14 +/- 2 ppm.