Vertical array records of the 1995 Kobe earthquake showed clearly nonlinearity of soil behavior. Although there have been few observational validations, earlier scientific publications on soil nonlinearity appeared since 1970's. In theoretical studies, nonlinearity has been introduced in constitutive relations as the quadratic correction to Hooke's law, but the question of the types of soil nonlinearity was not investigated. Here we show, on the basis of data processing and nonlinear system identification technique, that sediments represent physical systems possessing mostly odd types of nonlinearity. Even-order nonlinearities become comparable with odd-order ones only in special cases, like liquefied soils. Though the even and odd nonlinearities have much in common, the frequencies of combination harmonics, shapes of seismic solitary waves, and other nonlinear phenomena are different in these two cases. On the whole, the nonlinearity in soils leads to changes in spectra of propagating seismic signals: the spectra tending to take the form of E(f)~f--n.