  We prove the equivalence of two seemingly very different ways of generalising
Rademacher's theorem to metric measure spaces. One such generalisation is based
upon the notion of forming partial derivatives along a very rich structure of
Lipschitz curves in a way analogous to the differentiability theory of
Euclidean spaces. This approach to differentiability in this generality appears
here for the first time and by examining this structure further, we naturally
arrive to several descriptions of Lipschitz differentiability spaces.
