  We explore cosmological implications of a genuinely Weyl invariant (WI)
gravitational interaction. The latter reduces to general relativity in a
particular conformal frame for which the gravitational coupling and active
gravitational masses are fixed. Specifically, we consider a cosmological model
in this framework that is {\it dynamically} identical to the standard model
(SM) of cosmology. However, {\it kinematics} of test particles traveling in the
new background metric is modified thanks to a new (cosmological) fundamental
mass scale, $\gamma$, of the model. Since the lapse-function of the new metric
is radially-dependent any incoming photon experiences (gravitational)
red/blueshift in the {\it comoving} frame, unlike in the SM. Distance scales
are modified as well due to the scale $\gamma$. The claimed $4.4\sigma$ tension
level between the locally measured Hubble constant, $H_{0}$, with SH0ES and the
corresponding value inferred from the cosmic microwave background (CMB) could
then be significantly alleviated by an earlier-than-thought recombination.
Assuming vanishing spatial curvature, either one of the Planck 2018 (P18) or
dark energy survey (DES) yr1 data sets subject to the SH0ES prior imply that
$\gamma^{-1}$ is $O(100)$ times larger than the Hubble scale, $H_{0}^{-1}$.
Considering P18+SH0ES or P18+DES+SH0ES data set combinations, the odds against
vanishing $\gamma$ are over 1000:1 and 2000:1, respectively, and the model is
strongly favored over the SM with a deviance information criterion (DIC) gain
$\gtrsim 10$ and $\gtrsim 12$, respectively. The tension is reduced in this
model to $\sim 1.5$ and $1.3 \sigma$, respectively. We conclude that the
$H_{0}$ tension may simply result from a yet unrecognized fundamental symmetry
of the gravitational interaction -- Weyl invariance. (abridged)
