The determination of the mass composition of ultrahigh-energy cosmic rays remains one of the biggest challenges of astroparticle physics. We will show that the paradigm of shower universality can be applied to accurately reconstruct the properties of air showers, which includes information about the primary mass. The reconstruction is based solely on data from the surface detector of the Pierre Auger Observatory, which operates with a duty cycle of nearly 100 %. In contrast to purely empirical reconstruction methods, the foundation of the universality approach lies in physics models of the signal and arrival time distributions of secondary particles in air showers. In this contribution, results of the universality reconstruction are compared to their counterparts from the measurements of the Auger fluorescence detector. The focus of these comparisons is on the depth of the shower maximum, primary energy, and geometry. In addition, we extend these event-by-event comparisons to air shower simulations of various hadronic interaction models, primary masses, and energies. We will also present the performance of the method in estimating the muon content of air showers by comparing the true and reconstructed muon number for simulated showers.