The in-plane dispersion of the X X, Y states in a GaAs/AlAs “double-barrier structure” is measured by varying the angle of an in-plane magnetic field. When X X, Y (1) states in the emitter AlAs layer tunnel into collector XX,Y (m>1) states, a characteristic dumbbell shape is observed for the bias shift of the resonant tunneling peak versus magnetic field angle, with the major axis along [1 1 0] or [ 1̄ 1 0] . This corresponds to an elliptical constant energy surface in the collector AlAs layer which is rotated by 45° with respect to the bulk Fermi surface. We explain the new symmetry by X X-X Y interface band mixing which is closely analogous to the widely studied Γ-X Z mixing. Our results provide new insight into the microscopic origin of both types of mixing.