There are two concepts of time in special relativity (SR): The invariant evolution parameter is natural, but object-related, proper time τ. The time coordinate is man-made, coordinate time t. We present a novel, all-natural formulation of Euclidean relativity (ER): The invariant evolution parameter is natural, cosmic time θ = 1/Hθ (Hubble parameter Hθ). The time coordinate is natural τ measured by clocks. All objects move through 4D Euclidean space (ES) at the speed of light c. Each object experiences two orthogonal projections from absolute ES as a relative, Euclidean spacetime: the axis of its current 4D motion as proper time and the other three axes as proper space. ER and SR describe the same cosmos, but in different metrics. ER confirms the Lorentz factor of SR and—only locally—the gravitational time dilation of general relativity (GR). Thus, the predictions made by SR are exact, while either GR or ER is an approximation. GR is probably that approximation because absolute θ suits quantum mechanics better than relative t. τ is the length of a 4D Euclidean vector τ4D. ES diagrams do not show events, but τ4D. Gravity makes a comeback as a force. Any acceleration rotates an object’s τ4D and curves its worldline in ES. Information hidden in θ or τ4D is not available in SR/GR, but enables ER to predict time’s arrow, galactic motion, the Hubble tension, entanglement, the baryon asymmetry, and more. Remarkably, ER requires neither dark energy nor non-locality. Is ER the key to unifying physics?