Non-Hermitian systems containing gain or loss commonly host exceptional point degeneracies, not the diabolic points that, in Hermitian systems, play a key role in topological transitions and related phenomena. Non-Hermitian Hamiltonians with parity-time symmetry can have real spectra but generally nonorthogonal eigenstates, impeding the emergence of diabolic points. We introduce a pair of symmetries that induce not only real eigenvalues but also pairwise eigenstate orthogonality. This allows non-Hermitian systems to host Dirac points and other diabolic points. We construct non-Hermitian models exhibiting three exemplary phenomena previously limited to the Hermitian regime: Haldane-type topological phase transition, Landau levels without magnetic fields, and Weyl points. This establishes a new connection between non-Hermitian physics and the rich phenomenology of diabolic points.