This paper addresses, for the first time, a key aspect of the phenomenology of compact symmetric objects (CSOs): the characteristics of their radio spectra. We present a radio-spectrum description of a complete sample of high-luminosity CSOs (CSO-2s), which shows that they exhibit the complete range of spectral types, including flat-spectrum sources (alpha >= -0.5), steep-spectrum sources (alpha < -0.5), and peaked-spectrum sources. We show that there is no clear correlation between spectral type and size, but there is a correlation between the high-frequency spectral index and both object type and size. We also show that, to avoid biasing the data and to understand the various classes of active galactic nuclei (AGN) involved, the complete range of spectral types should be included in studying the general phenomenology of compact jetted AGN, and that complete samples must be used, selected over a wide range of frequencies. We discuss examples that demonstrate these points. We find that the high-frequency spectral indices of CSO-2s span -1.3 < alpha (hi) < -0.3 and hence that radio spectral signatures cannot be used to discriminate definitively between CSO-2s, binary galactic nuclei, and millilensed objects, unless they have alpha(hi) > -0.3.