It has been argued that Hořava gravity needs to be extended to include terms that mix spatial and time
derivatives in order to avoid unacceptable violations of Lorentz invariance in the matter sector. In an earlier
paper we have shown that including such mixed derivative terms generically leads to 4th instead of 6th
order dispersion relations and this could be (naïvely) interpreted as a threat to renormalizability. We have
also argued that power counting renormalizability is not actually compromised, but instead the simplest
power counting renormalizable model is not unitary. In this paper we consider the Lifshitz scalar as a toy
theory and we generalize our analysis to include higher order operators. We show that models which are
power counting renormalizable and unitary do exist. Our results suggest the existence of a new class of
theories that can be thought of as Hořava gravity with mixed derivative terms.