  We provide a full analysis of ghost free higher derivative field theories
with coupled degrees of freedom. Assuming the absence of gauge symmetries, we
derive the degeneracy conditions in order to evade the Ostrogradsky ghosts, and
analyze which (non)trivial classes of solutions this allows for. It is shown
explicitly how Lorentz invariance avoids the propagation of "half" degrees of
freedom. Moreover, for a large class of theories, we construct the field
redefinitions and/or (extended) contact transformations that put the theory in
a manifestly first order form. Finally, we identify which class of theories
cannot be brought to first order form by such transformations.
