This paper presents high-resolution X-ray observations with Chandra of NGC 4258 and infers the nature of the so-called in this galaxy. The anomalous arms dominate the X-ray image; diffuse X-ray emission from the plateau regions, seen in radio and Hα imaging, is also found. X-ray spectra have been obtained at various locations along the anomalous arms and are well described by thermal (MEKAL) models with kT in the range 0.37-0.6 keV. The previously known kiloparsec-scale radio jets are surrounded by cocoons of X-ray emitting gas for the first 350 pc of their length. The radio jets, seen in previous VLBA and VLA observations, propagate perpendicular to the compact nuclear gas disk (imaged in water vapor maser emission). The angle between the jets and the rotation axis of the galactic disk is 60°. The jets shock the normal interstellar gas along the first 350 pc of their length, causing the hot, X-ray emitting cocoons noted above. At a height of z = 175 pc from the disk plane, the jets exit the normal gas disk and then propagate though the low-density halo until they reach hot spots (at 870 pc and 1.7 kpc from the nucleus), which are seen in radio, optical line, and X-ray emission. These jets must drive mass motions into the low-density halo gas. This high-velocity halo gas impacts on the dense galactic gas disk and shock heats it along and around a of damage, which is the projection of the jets onto the galactic gas disk as viewed down the galaxy disk rotation axis. However, because NGC 4258 is highly inclined (i = 64°), the line of damage projects on the sky in a different direction from the jets themselves. We calculate the expected P.A. of the line of damage on the sky and find that it coincides with the anomalous arms to within 2°. It is therefore proposed that the anomalous arms, which are known to lie in the galactic disk, represent disk gas that has been shocked by mass motions driven by the out-of-plane radio jets. In the inner ( 1 kpc), some of the disk gas is blown out of the disk plane toward the halo on the opposite side of the disk from the relevant radio jet; this effect causes the arms to curve and is also responsible for the so-called plateau emission. An alternative is that the jet directions were different in the past, so they projected onto the disk in different directions. Our picture accounts for: (1) the diffuse character of the anomalous arms, (2) the inferred ionization of the optical line-emitting gas by shock waves, (3) the angle between the anomalous arms and the radio jets, (4) the sharp brightness gradients along the outer edges of the anomalous arms (these edges represent the standing shock where the high velocity, jet-driven halo gas meets the disk), (5) the existence of the plateaus, and (6) the wide range of radial velocities observed in the plateaus.