Imaging Fabry-Perot spectrophotometry has revealed a diffuse ionized medium (DIM) pervading the inner 10 kpc of the disk of NGC 1068. These data show that the DIM has high [N II] lambda-6584/H-alpha emission-line ratios (from 0.6 to 1.3) over much of the disk, with even higher ratios (from 1.3 to 2.5) near the nucleus and in isolated regions near the radio jet axis. Previous studies find that other low-excitation emission, such as [O II] lambda-3727 and [S II] lambda-lambda-6716, 6731, is also high away from the jet axis, while high-excitation emission such as [Ne v] lambda-lambda-3346, 3426 and He II lambda-4686 is found only within two diametric sectors of opening angle approximately 80-degrees, aligned along the axis. We show that these emission characteristics can be understood in the light of the developing picture of NGC 1068, in which its intrinsic type 1 Seyfert nucleus, concealed by an obscuring medium, is visible only through scattered, polarized light. In this picture the gas excitation is anisotropic, the high-excitation gas along the jet axis being photoionized by direct nuclear continuum, while the lower excitation gas away from the axis sees only indirect nuclear emission scattered into it. We consider in detail, a model of low optical depth (tau-S = 0.1), near-nuclear Thomson scattering and show that even with moderate dust extinction (E less-than-or-equal-to 0.3 mag kpc-1), this mechanism is sufficient to energize the DIM. This model provides ionization parameters that produce both globally enhanced, low-ionization lines such as [N II] lambda-6584, [O II] lambda-3727, and [S II] lambda-lambda-6716, 6731 and high-excitation emission such as [Ne v] lambda-lambda-3346, 3426 and He II lambda-4686 confined to the two diametric sectors along the radio axis. These Fabry-Perot data require a flux of ionizing photons far larger than those produced by dust-scattering models using standard grain compositions and size distributions. We therefore favor electrons as the dominant source of scattering of the nuclear ionizing radiation.