By examining the magnetic structure at sites in the bright coronal interiors of active regions that are not flaring but exhibit persistent strong coronal heating, we establish some new characteristics of the magnetic origins of this heating. We have examined the magnetic structure of these sites in five active regions, each of which was well observed by both the Yohkoh SXT and the Marshall Space Flight Center Vector Magnetograph and showed strong shear in its magnetic field along part of at least one neutral line (polarity inversion). Thus, we can assess whether this form of nonpotential field structure in active regions is a characteristic of the enhanced coronal heating and vice versa. From 27 orbits' worth of Yohkoh SXT images of the five active regions, we have obtained a sample of 94 persistently bright coronal features (bright in all images from a given orbit), 40 long (20,000 km) neutral-line segments having strong magnetic shear throughout (shear angle greater than 45°), and 39 long neutral-line segments having weak magnetic shear throughout (shear angle less than 45°). From this sample, we find that (1) all of our persistently bright coronal features are rooted in magnetic fields that are stronger than 150 G, (2) nearly all (95%) of these enhanced coronal features are rooted near neutral lines (closer than 10,000 km), (3) a great majority (80%) of the bright features are rooted near strong-shear portions of neutral lines, (4) a great majority (85%) of long strong-shear segments of neutral lines have persistently bright coronal features rooted near them, (5) a large minority (40%) of long weak-shear segments of neutral lines have persistently bright coronal features rooted near them, and (6) the brightness of a persistently bright coronal feature often changes greatly over a few hours. From these results, we conclude that most persistent enhanced heating of coronal loops in active regions (1) requires the presence of a polarity inversion in the magnetic field near at least one of the loop footpoints, (2) is greatly aided by the presence of strong shear in the core magnetic field along that neutral line, and (3) is controlled by some variable process that acts in this magnetic environment. We infer that this variable process is low-lying reconnection accompanying flux cancellation.