We propose UniPhy, a common latent-conditioned neural constitutive model that can encode the physical properties of diverse materials. At inference UniPhy allows `inverse simulation' i.e. inferring material properties by optimizing the scene-specific latent to match the available observations via differentiable simulation. In contrast to existing methods that treat such inference as system identification, UniPhy does not rely on user-specified material information. Compared to prior neural constitutive modeling approaches which learn scene specific networks, the shared training across materials improves both, robustness and accuracy of the estimates. We train UniPhy using simulated trajectories across diverse geometries and materials -- elastic, plasticine, sand, and fluids~(Newtonian & non-Newtonian). At inference, given an object with unknown material properties, UniPhy can infer the material properties via latent optimization to match the motion observations, and can then allow re-simulating the object under diverse scenarios. We compare UniPhy against prior inverse simulation methods, and show that the inference from UniPhy enables more accurate replay and re-simulation under novel conditions.