  Different extended objects can fall in different ways, depending on their
internal structures. Some motions are nevertheless impossible, regardless of
internal structure. This paper derives universal constraints on extended-body
motion, both in Newtonian gravity and in general relativity. In both theories,
we identify a weak notion of "local symmetry" which precludes certain force and
torque combinations. Local symmetries imply that certain components of a body's
quadrupole moment cannot affect its motion. They also imply that some forces
arise only in combination with appropriate torques. Many of these symmetries
are shown to be determined by the algebraic structure of the tidal tensor. In
general relativity, we thus relate qualitative features of extended-body motion
to the Petrov type of the spacetime. Doing so shows that local symmetries are
in fact ubiquitous. In general relativity, there are at least two in all
algebraically-special spacetimes. Some of these are generated by Killing
vectors and some by conformal Killing-Yano tensors. However, many local
symmetries do not fall into either of these classes.
