  We extend and explore the general non-relativistic effective theory of dark
matter (DM) direct detection. We describe the basic non-relativistic building
blocks of operators and discuss their symmetry properties, writing down all
Galilean-invariant operators up to quadratic order in momentum transfer arising
from exchange of particles of spin 1 or less. Any DM particle theory can be
translated into the coefficients of an effective operator and any effective
operator can be simply related to most general description of the nuclear
response. We find several operators which lead to novel nuclear responses.
These responses differ significantly from the standard minimal WIMP cases in
their relative coupling strengths to various elements, changing how the results
from different experiments should be compared against each other. Response
functions are evaluated for common DM targets - F, Na, Ge, I, and Xe - using
standard shell model techniques. We point out that each of the nuclear
responses is familiar from past studies of semi-leptonic electroweak
interactions, and thus potentially testable in weak interaction studies. We
provide tables of the full set of required matrix elements at finite momentum
transfer for a range of common elements, making a careful and fully
model-independent analysis possible. Finally, we discuss embedding
non-relativistic effective theory operators into UV models of dark matter.
