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We investigate the coupling of matter to geometry in conformal quadratic Weylgravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. Thecoupling explicitly satisfies the conformal invariance of the theory. Byexpressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of theWeyl scalar, the gravitational action can be linearized, leading in the Riemannspace to a conformally invariant $f\left(R,L_m\right)$ type theory, with thematter Lagrangian nonminimally coupled to the Ricci scalar. We obtain thegravitational field equations of the theory, as well as the energy-momentumbalance equations. The divergence of the matter energy-momentum tensor does notvanish, and an extra force, depending on the Weyl vector, and matter Lagrangianis generated. The thermodynamic interpretation of the theory is also discussed.The generalized Poisson equation is derived, and the Newtonian limit of theequations of motion is considered in detail. The perihelion precession of aplanet in the presence of an extra force is also considered, and constraints onthe magnitude of the Weyl vector in the Solar System are obtained from theobservational data of Mercury. The cosmological implications of the theory arealso considered for the case of a flat, homogeneous and isotropicFriedmann-Lemaitre-Robertson-Walker geometry, and it is shown that the modelcan give a good description of the observational data for the Hubble functionup to a redshift of the order of $z\approx 3$.

Coupling matter and curvature in Weyl geometry: conformally invariant $f\left(R,L_m\right)$ gravity