Interactions between biomolecules are characterized by where they occur and how they are organized, e.g., the alignment of lipid molecules to form a membrane. However, spatial and angular information are mixed within the image of a fluorescent molecule–the microscope’s dipole-spread function (DSF). We demonstrate the pixOL algorithm to simultaneously optimize all pixels within a phase mask to produce an engineered Green’s tensor–the dipole extension of point-spread function engineering. The pixOL DSF achieves optimal precision to simultaneously measure the 3D orientation and 3D location of a single molecule, i.e., 4.1◦ orientation, 0.44 sr wobble angle, 23.2 nm lateral localization, and 19.5 nm axial localization precisions in simulations over a 700 nm depth range using 2500 detected photons. The pixOL microscope accurately and precisely resolves the 3D positions and 3D orientations of Nile red within a spherical supported lipid bilayer, resolving both membrane defects and differences in cholesterol concentration in six dimensions.