We have mapped NGC 3718, a nearby bright galaxy in a loose group, and its companion NGC 3729 in the 21 cm line of neutral hydrogen. NGC 3718 is a strikingly unusual galaxy with a strong straight dust lane across the center, peculiar diffuse spiral arms, and an extended disk of neutral hydrogen. Earlier work showed the gas disk to be strongly twisted, warping through edge-on where we see the straight dust lane; stars formed in this gas appear to make up the "spiral arms." Our improved maps show a twisted but bisymmetric disk of gas extending to 7' or 35 kpc, where the orbital period is roughly 1 Gyr. It is surrounded by fragmentary spiral features, and a streamer of gas extending to a cloud lying 12' or 60 kpc to the north. We use inspector, a task in GIPSY, to fit a tilted-ring model interactively to slices through the H I data cube. The apparent major axis swings through 100. from the innermost gas orbits at 30 '' radius to the outer edge. When viewed in the reference frame of the galaxy's stellar disk, the innermost gas orbits are nearly polar, while the outer rings of gas are tilted at 30 degrees-40 degrees. The line of nodes, where the gas orbits pass through the plane of the stellar disk, twists by roughly 90. about the pole. We do not see gas orbiting in the plane of the stellar disk. If we assume that the galaxy's dark halo shares the same midplane, then the observed twist can be explained by differential precession in a dynamical model in which the dark halo is fairly round. The run of tilt with radius is close to what is required for the warped gas disk to precess rigidly in the galaxy's gravitational field without changing its shape. This fact probably accounts for the longevity of the twisted structure.