  We study the slicing and fine properties of functions in
$\mathrm{BV}^{\mathcal A}$, the space of functions with bounded $\mathcal
A$-variation. Here, $\mathcal A$ is a homogeneous linear differential operator
with constant coefficients (of arbitrary order). Our main result is the
characterization of all $\mathcal A$ satisfying the following one-dimensional
structure theorem: every $u \in \mathrm{BV}^{\mathcal A}$ can be sliced into
one-dimensional $\mathrm{BV}$-sections. Moreover, decomposing $\mathcal A u$
into an absolutely continuous part $\mathcal A^a u$, a Cantor part $\mathcal
A^c u$ and a jump part $\mathcal A^j u$, each of these measures can be
recovered from the corresponding classical $D^a,D^c$ and $D^j$ $BV$-derivatives
of its one-dimensional sections. By means of this result, we are able to
analyze the set of Lebesgue points as well as the set of jump points where
these functions have approximate one-sided limits. Thus, proving a structure
and fine properties theorem in $\mathrm{BV}^{\mathcal A}$. Our results extend
most of the classical fine properties of $\mathrm{BV}$ (and all of those known
for $\mathrm{BD}$). In particular, we establish a slicing theory and fine
properties for $\mathscr {BV}^k, \mathrm{BD}^k$ and a whole class of
$\mathrm{BV}^{\mathcal A}$-spaces that is not covered by the existing theory.
