  We compute conformally covariant actions and operators for tensors with mixed
symmetries in arbitrary dimension $d$. Our results complete the classification
of conformal actions that are quadratic on arbitrary tensors with three
indices, which allows to write corresponding conformal actions for all tensor
species that appear in the decomposition of the distorsion tensor of an
arbitrary metric-affine theory of gravity including both torsion and
nonmetricity. We also discuss the degrees of freedom that such theories are
propagating, as well as interacting metric-affine theories that enjoy the
conformal actions in the Gaussian limit.
