A major impediment towards realizing technologies based on the emerging principles of antiferromagnetic spintronics is the shortage of suitable materials. In this paper, we propose a design of polar|nonpolar heterostructures of perovskite oxides, where a single unit cell of SrIrO3 is sandwiched between a thin film of LaAlO3 and a substrate of SrTiO3. Our calculations within the framework of density-functional theory + Hubbard U + spin-orbit coupling reveal a two-dimensional conducting layer with electron and hole pockets at the interface, exhibiting a strong anisotropic Rashba-Dresselhaus effect along with noncollinear antiferromagnetism, indicating the possibility of realizing a spin-orbit torque. An insightful physical model for the anisotropic Rashba-Dresselhaus effect nicely interprets our results, providing an estimate for the Rashba-Dresselhaus coefficients and illustrating pseudospin orientation.We also observe a proximity-induced prominent Rashba-like effect for Ti 3d empty bands. Our results suggest that the heterostructure may possess the essential ingredients for antiferromagnetic spintronics, deserving experimental verification.