  We study substructures of the Weyl group of conformal transformations of the
metric of (pseudo)Riemannian manifolds. These substructures are identified by
differential constraints on the conformal factors of the transformations which
are chosen such that their composition is associative. Mathematically, apart
from rare exceptions, they are partial associative groupoids, not groups, so
they do not have an algebra of infinitesimal transformations, but this
limitation can be partially circumvented using some of their properties
cleverly. We classify and discuss the substructures with two-derivatives
differential constraints, the most famous of which being known as the harmonic
or restricted Weyl group in the physics literature, but we also show the
existence of a lightcone constraint which realizes a proper subgroup of the
Weyl group. We then show the physical implications that come from invariance
under the two most important substructures, concentrating on classical
properties of the energy-momentum tensor and a generalization of the quantum
trace anomaly. We also elaborate further on the harmonic substructure, which
can be interpreted as partial gauge fixing of full Weyl invariance using BRST
methods. Finally, we discuss how to construct differential constraints of
arbitrary higher-derivative order and present, as examples, generalizations
involving scalar constraints with four and six derivatives.
