  Linearized supergravity in arbitrary dimension is reformulated into a first
order formalism which treats the graviton and its dual on the same footing at
the level of the action. This generalizes previous work by other authors in two
directions: 1) we work in arbitrary space-time dimension, and 2) the gravitino
field and supersymmetry are also considered. This requires the construction of
conformally invariant curvatures (the Cotton fields) for a family of mixed
symmetry tensors and tensor-spinors, whose properties we prove (invariance;
completeness; conformal Poincar\'e lemma). We use these geometric tools to
solve the Hamiltonian constraints appearing in the first order formalism of the
graviton and gravitino: the constraints are solved through the introduction of
prepotentials enjoying (linearized) conformal invariance. These new variables
(two tensor fields for the graviton, one tensor-spinor for the gravitino) are
injected into the action and equations of motion, which take a geometrically
simple form in terms of the Cotton tensor(-spinors) of the prepotentials. In
particular, the equations of motion of the graviton are equivalent to twisted
self-duality conditions. We express the supersymmetric transformations of the
graviton and gravitino into each other in terms of the prepotentials. We also
reproduce the dimensional reduction of supergravity within the prepotential
formalism. Finally, our formulas in dimension five are recovered from the
dimensional reduction of the already known prepotential formulation of the
six-dimensional $\mathcal{N}=(4,0)$ maximally supersymmetric theory.
