The radiation characteristics of diagonal horns are investigated by means of Gaussian-Hermite modes. It is shown that, for reasonably long horns, the beamwidths in the principal and 45-degrees planes are equal to within 10%, and all sidelobes are below -15 dB. It is also demonstrated that the phase center of a diffraction-limited horn is close to the aperture, whereas the phase center of a constant-beamwidth horn is behind the throat. The maximum coupling to the lowest order co-polar Gaussian mode is 84%, and the total amount of power coupled into the cross-polarized lobes is 9.5%. More significantly, the aperture efficiency of a Cassegrain antenna fed by a diagonal horn has a maximum value of 81%, which compares with 87% for a corrugated horn. The maximum efficiency is achieved when the aperture of a diffraction-limited horn is placed at a confocal tertiary focus, although a secondary focus gives an aperture efficiency that is only 10% lower, suggesting that diagonal horns are suitable for focal-plane arrays.