Since the discovery of the unusual prototype SN 2002cx, the eponymous class of low-velocity, hydrogen-poor supernovae has grown to include at most another two dozen members identified from several heterogeneous surveys, in some cases ambiguously. Here we present the results of a systematic study of 1077 hydrogen-poor supernovae discovered by the Palomar Transient Factory, leading to nine new members of this peculiar class. Moreover we find there are two distinct subclasses based on their spectroscopic, photometric, and host galaxy properties: The "SN 2002cx-like" supernovae tend to be in later-type or more irregular hosts, have more varied and generally dimmer luminosities, have longer rise times, and lack a Ti II trough when compared to the \SN 2002es-like" supernovae. None of our objects show helium, and we counter a previous claim of two such events. We also find that these transients comprise 5.6+17 -3:7% (90% confidence) of all SNe Ia, lower compared to earlier estimates. Combining our objects with the literature sample, we propose that these subclasses have two distinct physical origins.