  In the first part, we discuss the interplay between local scale invariance
and metric-affine degrees of freedom from few distinct points of view. We
argue, rather generally, that the gauging of Weyl symmetry is a natural
byproduct of requiring that scale invariance is a symmetry of a gravitational
theory that is based on a metric and on an independent affine structure degrees
of freedom. In the second part, we compute the N\"other identities associated
with all the gauge symmetries, including Weyl, Lorentz and diffeomorphisms
invariances, for general actions with matter degrees of freedom, exploiting a
gauge covariant generalization of the Lie derivative. We find two equivalent
ways to approach the problem, based on how we regard the spin-connection
degrees of freedom, either as an independent object or as the sum of two Weyl
invariant terms. The latter approach, which rests upon the use of a new
connection, denoted $\hat{\nabla}$, is particularly convenient and constitutes
one of our main results.
