  We demonstrate that ultracold symmetric top molecules loaded into an optical
lattice can realize highly tunable and unconventional models of quantum
magnetism, such as an XYZ Heisenberg spin model. We show that anisotropic
dipole-dipole interactions between molecules can lead to effective spin-spin
interactions which exchange spin and orbital angular momentum. This exchange
produces effective spin models which do not conserve magnetization and feature
tunable degrees of spatial and spin-coupling anisotropy. In addition to
deriving pure spin models when molecules are pinned in a deep optical lattice,
we show that models of itinerant magnetism are possible when molecules can
tunnel through the lattice. Additionally, we demonstrate rich tunability of the
effective models' parameters using only a single microwave frequency, in
contrast to proposals with $^1\Sigma$ diatomic molecules, which often require
many microwave frequencies. Our results are germane not only for experiments
with polyatomic symmetric top molecules, such as methyl fluoride (CH$_3$F), but
also diatomic molecules with an effective symmetric top structure, such as the
hydroxyl radical OH.
