One can link the type I to types II and III in the following way: if the cumulative distribution function of some random variable X is of type II, and with the positive numbers as support, i.e. F(x; 0, \sigma, \alpha), then the cumulative distribution function of \ln X is of type I, namely F(x; \ln \sigma, 1/\alpha, 0). Similarly, if the cumulative distribution function of X is of type III, and with the negative numbers as support, i.e. F(x; 0, \sigma, -\alpha), then the cumulative distribution function of \ln (-X) is of type I, namely F(x; -\ln \sigma, 1/\alpha, 0).
