  It has been argued that Horava gravity needs to be extended to include terms
that mix spatial and time derivatives in order 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 (naively) 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 Horava gravity with mixed derivative terms.
