  For scalar conservation laws driven by a rough path $z(t)$, in the sense of
Lions, Perthame and Souganidis in arXiv:1309.1931, we show that it is possible
to replace $z(t)$ by a piecewise linear path, and still obtain the same
solution at a given time, under the assumption of a convex flux function in one
spatial dimension. This result is connected to the spatial regularity of
solutions. We show that solutions are spatially Lipschitz continuous for a
given set of times, depending on the path and the initial data. Fine properties
of the map $z \mapsto u(\tau)$, for a fixed time $\tau$, are studied. We
provide a detailed description of the properties of the rough path $z(t)$ that
influences the solution. This description is extracted by a "factorization" of
the solution operator (at time $\tau$). In a companion paper, we make use of
the observations herein to construct computationally efficient numerical
methods.
