  We prove uniqueness and existence theorems for four-dimensional
asymptotically flat, Ricci-flat, gravitational instantons with a torus
symmetry. In particular, we prove that such instantons are uniquely
characterised by their rod structure, which is data that encodes the fixed
point sets of the torus action. Furthermore, we establish that for every
admissible rod structure there exists an instanton that is smooth up to
possible conical singularities at the axes of symmetry. The proofs involve
adapting the methods that are used to establish black hole uniqueness theorems,
to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where
the target space is directly related to the metric (rather than auxiliary
potentials). We also give an elementary proof of the nonexistence of
asymptotically flat toric half-flat instantons. Finally, we derive a general
set of identities that relate asymptotic invariants such as the mass to the rod
structure.
