  The general second-order massive field equations for arbitrary positive
integer spin in three spacetime dimensions, and their "self-dual" limit to
first-order equations, are shown to be equivalent to gauge-invariant
higher-derivative field equations. We recover most known equivalences for spins
1 and 2, and find some new ones. In particular, we find a non-unitary massive
3D gravity theory with a 5th order term obtained by contraction of the Ricci
and Cotton tensors; this term is part of an N=2 super-invariant that includes
the "extended Chern-Simons" term of 3D electrodynamics. We also find a new
unitary 6th order gauge theory for "self-dual" spin 3.
