The importance of hyperfine structure observed in molecular beam or high-resolution microwave spectroscopy experiments has been almost completely overlooked by NMR spectroscopists and theoreticians. In the present work, we show for a series of diatomic molecules that the indirect spin-spin coupling tensor, of fundamental importance to magnetic resonance spectroscopy, is completely characterized by the hyperfine measurements. The hyperfine parameter c 4 is known to be equivalent to the isotropic spin-spin coupling constant, J iso ; what has not been exploited is the relationship between c 3 and the anisotropic portion of the spin-spin coupling tensor, ∆J. Through comparisons to highly precise experimental data available for LiH, LiF, KF, Na 2 , and ClF, multiconfigurational SCF calculations using balanced complete active spaces and large correlation-consistent basis sets have been employed to establish the reliability of such calculations for determining the complete tensor rather than simply J iso . The experimental data are for "isolated" molecules, making them ideal for comparison with ab initio results; agreement is generally within a few percent after accounting for rovibrational effects. These results, combined with further calculations on a larger set of diatomic molecules (HF, BF, AlF, KNa, HCl, NaF), provide new insights into the nature of indirect spin-spin coupling. Calculations indicate the importance of each of the various coupling mechanisms. The influence of the Fermicontact mechanism, traditionally thought to be the dominant contribution to J iso , is shown to vary considerably even for couplings between first-row elements. General conclusions about the relative importance of all mechanisms to both the isotropic and anisotropic portions of the coupling tensor are discussed, and periodic trends are proposed.