This paper presents high resolution X-ray observations with Chandra of NGC 4258 and infers the nature of the so called ``anomalous arms'' in this galaxy. The anomalous arms dominate the X-ray image; diffuse X-ray emission from the ``plateaux'' 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 kpc-scale radio jets are surrounded by cocoons of hot 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 ``line 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 to 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∘. (Abstract truncated). Comment: 12 pages plus 9 figures, to be published in the Astrophysical Journal, v560, nr 1, pt 1 (Oct 10, 2001 issue
