  In this paper we revisit the motivation and construction of a unified theory
of gravity and electromagnetism, following Weyl's insights regarding the
appealing potential connection between the gauge invariance of electromagnetism
and the conformal invariance of the gravitational field. We highlight that
changing the local symmetry group of spacetime permits to construct a theory in
which these two symmetries are combined into a putative gauge symmetry but with
second-order field equations and non-trivial mass scales, unlike the original
higher-order construction by Weyl. We prove that the gravitational field
equations are equivalent to the (trace-free) Einstein field equations, ensuring
their compatibility with known tests of general relativity. As a corollary, the
effective cosmological constant is rendered radiatively stable due to Weyl
invariance. A novel phenomenological consequence characteristic of this
construction, potentially relevant for cosmological observations, is the
existence of an energy scale below which effects associated with the
non-integrability of spacetime distances, and an effective mass for the
electromagnetic field, appear simultaneously (as dual manifestations of the use
of Weyl connections). We explain how former criticisms against Weyl's ideas
lose most of their power in its present reincarnation, which we refer to as
Weyl relativity, as it represents a Weyl-invariant, unified description of both
the Einstein and Maxwell field equations.
