Elementary chemistry distinguishes two kinds of strong bonds between atoms in molecules: the covalent bond, where bonding arises from valence electron pairs shared between neighboring atoms, and the ionic bond, where transfer of electrons from one atom to another leads to Coulombic attraction between the resulting ions. We present a third, distinct bonding mechanism: perpendicular paramagnetic bonding, generated by the stabilization of antibonding orbitals in their perpendicular orientation relative to an external magnetic field. In strong fields such as those present in the atmospheres of white dwarfs (on the order of 10(5) teslas) and other stellar objects, our calculations suggest that this mechanism underlies the strong bonding of H-2 in the (3)Sigma(+)(u)(1 sigma(g)1 sigma*(u)) triplet state and of He-2 in the (1)Sigma(+)(g)(1 sigma(2)(g)1 sigma*(2)(u)) singlet state, as well as their preferred perpendicular orientation in the external field.