We report the results of an imaging survey of 2.122 mu m H-2 emission from similar to 60 planetary nebulae (PNs). We detected 23 PNs, including 11 first-time detections. Adding from the literature 48 PNs previously searched for H-2, we find that similar to 40% of the combined sample displays H-2 emission. The detections are dominated by objects that possess bipolar morphology (i.e., those that display well-defined polar lobes and/or equatorial regions in optical images). All seven H-2-emitting PNs that are not considered bipolar (based on H alpha images) are rings or disks, which strongly suggests that these objects are in fact bipolar PNs viewed at large inclinations. This interpretation also follows from the fact that the brightest H-2 emission is typically located along the equatorial plane (waist) of a bipolar nebula and that several nebulae that are bipolar in optical emission lines appear ringlike in H-2. We therefore confirm the validity of Gatley's rule: the detection of H-2 emission from a PN confirms the bipolar structure of that PN. The H-2-bright waist (or torus, in the case of a Ring-like PN) is presumably the remnant of a molecule-rich circumstellar disk that predates the production of a PN. We confirm and strengthen the claim that PNs detected in H-2 are confined to lower Galactic latitudes, and therefore had larger progenitor masses, than PNs without detectable emission. These results therefore reinforce the connection between PN progenitor mass, PN morphology, and the presence or absence of 2.122 mu m H-2 emission.