Today’s physics describes nature in “empirical concepts” (based on observation). Examples are coordinate space/coordinate time in special relativity (SR), curved spacetime in general relativity (GR), and concepts of objects (particles, matter waves, photons, electromagnetic waves). Here we show: There is a complementary description that does not interfere with SR/GR. Euclidean relativity (ER) describes nature in “natural concepts” (immanent in all objects). Examples are proper space/proper time, curved worldlines in 4D Euclidean space (ES), and “wavematters” (pure energy). An object’s proper space d1, d2, d3 and proper time τ span its reference frame d1, d2, d3, d4 in ES (d4 = cτ). The orientation of its reference frame in absolute ES is relative. All energy moves through ES at the speed of light c. Absolute, cosmic time is the total distance covered in ES divided by c. Each object experiences its 4D motion as proper time. There is a 4D vector “flow of proper time” τ for each object. Any acceleration rotates an object’s τ and curves its worldline in flat ES. The 4D vector τ is crucial for objects that are very far away or entangled. These objects must be described in natural concepts. Information hidden in τ is not available in SR/GR. ER solves fundamental riddles, such as the nature of time, the Hubble tension, the wave–particle duality, and the baryon asymmetry. In ER, cosmic inflation, expanding space, dark energy, and non-locality are obsolete concepts.