The heavy-hole (HH) and light-hole (LH) components of the valence states in 3D bulk semiconductors can mix quantum mechanically as the dimensionality is reduced in forming 2D, 1D nanostructures and 0D quantum dots (QDs). This coupling controls the tuning of the excitonic fine structure splitting, provides an efficient channel for the spin coherence, and leads to polarization anisotropy of light emission, central to several quantum information schemes. Current understanding is that the mixing scales with the square of δV HL /∆ HL , where δV HL and ∆ HL are the coupling matrix elements of the crystal potential and the energy separation between the primary HH0 and LH0 states, respectively. We discuss two classes of HH-LH coupling mechanisms.First, coupling factors occurring through the numerator δV HL , referred to as "direct coupling", including the well-known (1) quantum confinement, (2) built-in strain, and (3) shape elongation, as well as three additional direct coupling mechanisms discussed here: (4) the intrinsic C 2v crystal field effect, (5) the local symmetry of the interface, and (6) the alloy disorder. We quantify these 6 direct HH-LH coupling effects by performing atomistic pseudopotential calculations on a range of strained and unstrained QDs of different morphologies. We find that in unstrained self-assembled QDs such as GaAs/AlGaAs effects (1)-(6) contribute 0%, 0%, 0%, 0%, 40%, and 60%, respectively, whereas in strained self-assembled QDs such as InGaAs/GaAs they contribute 0%, 0%, 78%, 0%, 8%, and 14%, respectively, to the direct HH-LH coupling δV HL . These relative contributions to direct HH-LH coupling differ significantly from what was previously believed.Second, we discover an unexpected HH-LH coupling that effectively reduces the denominator ∆ HL by the presence of a dense ladder of intermediate states between the HH0 and LH0 states (analogous to super-exchange in magnetism). Supercoupling amplifies and propagates the HH-LH interaction and is the dominant source of HH-LH mixing in strained nanostructures where ∆ HL is fairly large, so by the direct coupling mechanism alone δV HL /∆ HL would be expected to be rather small. Supercoupling explains a number of outstanding puzzles including the surprising fact that in strained (InAs/GaAs) QDs the mixing is very strong despite the fact that ∆ HL is large, and offers a new way to manipulate HH-LH mixing and hence associated properties in nanostructures.