The notion of odd and even operators introduced by Schr\"odinger is generalized and developed in this paper. It is shown that, accepting the hypothesis of Schr\"odinger that "only even operators exist" the undesirable negative energy solutions of the Dirac equation are eliminated since they do not combine with the observed states, while at the same time the relativistically invariant form of the Dirac equation can be retained. The fine structure formula for the energy levels remains exact and not merely approximate. However, it is proposed to relinquish these apparent advantages in order to substitute for Schr\"odinger's hypothesis one easier to accept but practically equivalent. This is that the fundamental potential between charges is an even operator. Possible ways of forming even operators out of classical mixed ones are discussed. The decomposition of the fundamental operators, ${\ensuremath{\alpha}}_{i}$ the "spin" operator, and ${x}_{i}$ the coordinate operator, indicates a remarkably simple form for the odd and even parts. The even parts prove to be immediately connected with observable properties of the electron. This leads one to believe that the division into odd and even parts is a useful performance aside from its application to the question of negative energy solutions. While the positive to negative transitions are made to vanish identically, the direct transitions between positive states remain quite unaffected. The "Klein difficulty" is examined and it proves to disappear with the negative energy states. Possible experimental tests of the new theory are shown to exist. The values of hyperfine structure separations and also the x-ray levels of heavy atoms should depend markedly on whether the Coulomb potential or just its even part acts on the electron. Some interesting possibilities suggested by the theory are mentioned at the end of the paper.