  Processes having the same bridges as a given reference Markov process
constitute its {\it reciprocal class}. In this paper we study the reciprocal
class of compound Poisson processes whose jumps belong to a finite set
$\mathcal{A} \subset \mathbb{R}^d$. We propose a characterization of the
reciprocal class as the unique set of probability measures on which a family of
time and space transformations induces the same density, expressed in terms of
the \textit{reciprocal invariants}. The geometry of $\mathcal{A}$ plays a
crucial role in the design of the transformations, and we use tools from
discrete geometry to obtain an optimal characterization. We deduce explicit
conditions for two Markov jump processes to belong to the same class. Finally,
we provide a natural interpretation of the invariants as short-time asymptotics
for the probability that the reference process makes a cycle around its current
state.
