  We introduce two general non-parametric methods for recovering paths of the
Brownian and jump components from high-frequency observations of a L\'evy
process. The first procedure relies on reordering of independently sampled
normal increments and thus avoids tuning parameters. The functionality of this
method is a consequence of the small time predominance of the Brownian
component, the presence of exchangeable structures, and fast convergence of
normal empirical quantile functions. The second procedure amounts to filtering
the increments and compensating with the final value. It requires a carefully
chosen threshold, in which case both methods yield the same rate of
convergence. This rate depends on the small-jump activity and is given in terms
of the Blumenthal-Getoor index. Finally, we discuss possible extensions,
including the multidimensional case, and provide numerical illustrations.
