We propose that the state of the Universe does not spontaneously violate CPT. Instead, the Universe after the big bang is the CPT image of the Universe before it, both classically and quantum mechanically. The pre- and postbang epochs comprise a universe-antiuniverse pair, emerging from nothing directly into a hot, radiation-dominated era. CPT symmetry selects a unique QFT vacuum state on such a spacetime, providing a new interpretation of the cosmological baryon asymmetry, as well as a remarkably economical explanation for the cosmological dark matter. Requiring only the standard three-generation model of particle physics (with right-handed neutrinos), a Z2 symmetry suffices to render one of the right-handed neutrinos stable. We calculate its abundance from first principles: matching the observed dark matter density requires its mass to be 4.8×108 GeV. Several other testable predictions follow: (i) the three light neutrinos are Majorana particles and allow neutrinoless double β decay; (ii) the lightest neutrino is massless; and (iii) there are no primordial long-wavelength gravitational waves. We mention connections to the strong CP problem and the arrow of time.