We explore cosmological implications of agenuinely Weyl invariant (WI) gravitational interaction.The latter reduces to general relativity in a particularconformal frame for which the gravitational coupling andactive gravitational masses are fixed.Specifically, we consider a cosmological model in thisframework that is dynamically identical to thestandard model (SM) of cosmology.However, kinematics of test particles travelingin the new background metric is modifiedthanks to a new (cosmological) fundamentalmass scale, γ, of the model that emergesas an integration constant of the classical field equations.Since the lapse-function of the new metric isradially-dependent any incoming photon experiences (gravitational)red/blueshift in the comoving frame, unlike in the SM.Distance scales are modified as well due to the scale γ.The claimed 4.4σ tension level between the locally measuredHubble constant, H 0, with SH0ES and the correspondingvalue inferred from the cosmic microwave background (CMB)could then be significantly alleviated byan earlier-than-thought recombination. Assumingvanishing spatial curvature, either one ofthe Planck 2018 (P18) or dark energy survey (DES)yr1 data sets subject to the SH0ES priorimply that γ -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 γ are over 1000:1 and 2000:1, respectively,and the model is strongly favored over the SM with adeviance information criterion (DIC) gain ≳ 10and ≳ 12, respectively.The H 0 tension is reduced in this modelto ∼ 1.5 and 1.3 σ, respectively.Allowing for a non-vanishing spatial curvature, γ -1halves to O(50) times H 0 -1.The capacity of two other major cosmologicalprobes, baryonic oscillations and type Ia supernovae,SNIa, to distinguish between the models is alsodiscussed. We conclude that the H 0 tension maysimply result from a yet unrecognized fundamentalsymmetry of the gravitational interaction — Weyl invariance.