We present spectra obtained with the Infrared Spectrograph on the Spitzer Space Telescope of 33 K giants and 20 A dwarfs to assess their suitability as spectrophotometric standard stars. The K giants confirm previous findings that the strength of the SiO absorption band at 8 mum increases for both later optical spectral classes and redder (B-V)₀ colors, but with considerable scatter. For K giants, the synthetic spectra underpredict the strengths of the molecular bands from SiO and OH. For these reasons, the assumed true spectra for K giants should be based on the assumption that molecular band strengths in the infrared can be predicted accurately from neither optical spectral class or color nor synthetric spectra. The OH bands in K giants grow stronger with cooler stellar temperatures, and they are stronger than predicted by synthetic spectra. As a group, A dwarfs are better behaved and more predictable than the K giants, but they are more likely to show red excesses from debris disks. No suitable A dwarfs were located in parts of the sky continuously observable from Spitzer, and with previous means of estimating the true spectra of K giants ruled out, it was necessary to use models of A dwarfs to calibrate spectra of K giants from observed spectral ratios of the two groups and then use the calibrated K giants as standards for the full database of infrared spectra from Spitzer. We also describe a lingering artifact that affects the spectra of faint blue sources at 24 mum.