Experiments are described which show that grating anomalies are not confined to that part of the incident light which is polarized with its electric vector perpendicular to the rulings. The appearance of the new anomalies is attributed to groove depth. The parallel anomalies have been found only when using gratings with grooves much deeper than the wavelength, and they vanish when the wavelength becomes comparable to the groove depth. It is shown why neither Strong nor Ingersoll found parallel anomalies, and why Rayleigh’s theory concludes that they should not exist. New experiments with thin dielectric coats show that very thin layers of dielectric of the order of a fraction of a wavelength thick modify the appearance of both parallel and perpendicular anomalies, although a thickness of about one and one-half wavelengths is required before the anomalous wavelength begins to shift. With thick layers of dielectric, both the parallel and perpendicular anomalies are shifted a calculable amount which depends on the refractive index.