The first-principles all electron relativistic calculations within the general gradient approximation are performed to investigate the interface structure, the electronic and the optical absorption properties of quaternary InAs/GaSb superlattices with InSb or GaAs type of interface. Because of the complexity and low symmetry of the quaternary interfaces, the equilibrium structural parameters of relaxed interfaces are determined by the minimization of total electronic energy and strain in InAs/GaSb superlattices. The band structures and the optical absorption spectra of InAs/GaSb superlattices with special InSb or GaAs and normal (two types are alternate) interfaces are calculated, with the consideration of the superlattice interface atomic relaxation effects. The calculation of relativistic Hartree-Fock functional and local density approximation with the plane wave method is also implemented to demonstrate the calculated band structure results. The calculated band structures of InAs/GaSb superlattices with different types of interfaces are systematically compared. We find that the chemical bonding and ionicity of interfacial Sb atoms are essentially important in determining the interface structures, the band structures and the optical properties of InAs/GaSb superlattices.