In Newtonian physics, momentum and energy are often treated as distinct entities, which happen to be separately conserved. In relativity, energy is regarded as the "time" component of the four-momentum. Energy and momentum are still separately conserved, but they mix under Lorentz transformations in a way that identifies energy with time and momentum with space.
Meanwhile, there is an analogous relationship energy:time::momentum:space in Newtonian physics, which becomes apparent in the Hamiltonian formulation. The Hamiltonian (energy) is the generator of time translations, whereas momentum is the generator of space translations.
What's the "secret" behind this? As pointed out in the comments, Noether's theorem relates energy and time via time translation symmetry in Newtonian mechanics, so this symmetry is probably also at play in relativity. But how? Is there a clear line of reasoning that goes from Noether's theorem + spacetime translation symmetry in relativity to "energy and momentum combine as a four-vector"?