Families of self-inverse functions and dilogarithm identities

We introduce a self-inverse function via an integral equivalent to a two-term combination of dilogarithms. We refer to this function as a fundamental form, since there is a family of extensions of this function that satisfy similar self-inverse and symmetric properties. We also construct a family of functions generalizing the fundamental form via two auxiliary parameters, which we refer to as shape and scale factors. Through new integration techniques, we introduce and prove a variety of dilogarithm identities and evaluations for dilogarithm ladders and for two-term dilogarithm combinations. The functions$\gemini_{a}^{b}(x)$ we introduce are referred to as gemini functions and may be seen as providing a broad framework in the derivation of and application of dilogarithm identities.