The polylogarithm of order n is defined by \(Li_ n(z)=\sum_{r}z^ r/r^ n\). This paper continues the work of the second author [ibid. 19, 345-373 (1984; Zbl 0556.12001)] on the strange identities between values of polylogarithms at certain algebraic points. A typical example for the dilogarithm is Watson's formula, \(Li_ 2(\alpha)-Li_ 2(\alpha^ 2)=\pi^ 2/42+\log^ 2\alpha\), where \(\alpha =\sec (2\pi /7)\) satisfies \(\alpha^ 3+2\alpha^ 2-\alpha -1=0\). Lewin has analysed the structure of these identities and the authors have used this analysis to discover identities for higher order polylogarithms. The interesting examples are all empirical, although the numerical verification leaves no doubt about their validity. The strange feature of most of these identities is that they do not seem to follow from the standard functional equations for the polylogarithm. In fact, they point to the existence of new types of functional equations, some of which are starting to emerge in further work by Lewin.