We introduce the Hartley Neural Operator (HNO), the exact real-valued counterpart of the Fourier Neural Operator (FNO). For real-valued solutions the complex FFT is redundant under conjugate symmetry, so HNO replaces it with the real Discrete Hartley Transform and learns one real multiplier per retained mode. Because the Hartley spectrum is not halved by conjugate symmetry, HNO keeps twice as many frequency corners as FNO, but each uses a single real weight where FNO uses a complex pair; the two operators are therefore iso-parametric at equal width and differ only in spectral basis.

Our central thesis is that the best basis is a property of the operator itself. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that a real Hartley multiplier diagonalizes exactly, favoring HNO. Time-dependent operators carry phase---oscillation in the wave equation, transport in advection, Burgers, and Navier--Stokes---which a real diagonal multiplier structurally cannot represent, favoring FNO by a margin that grows with phase content; the phaseless heat equation is the borderline case.

Training both operators identically across seven PDEs, three initial-condition families, and periodic and Dirichlet boundaries, we observe exactly this split, monotonic in operator phase content and consistent with the Green's-function theory we develop. The result is a predictive rule rather than a universal winner: match the spectral basis to the symmetry of the solution operator.