We develop the structure theory for transformations of weakly geometric rough
paths of bounded $1 < p$-variation and their controlled paths. Our approach
differs from existing approaches as it does not rely on smooth approximations.
We derive an explicit combinatorial expression for the rough path lift of a
controlled path, and use it to obtain fundamental identities such as the
associativity of the rough integral, the adjunction between pushforwards and
pullbacks, and a change of variables formula for rough differential equations
(RDEs). As applications we define rough paths, rough integration and RDEs on
manifolds, extending the results of [CDL15] to the case of arbitrary $p$.