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, Delta J. Through comparisons to highly precise experimental data available for LiH, LiF, KF, Na-2, and CIF, 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 Fermi-contact 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.