Recent Transition Region and Coronal Explorer (TRACE) observations have detected a class of active region loops whose physical properties are inconsistent with previous hydrostatic loop models. In this Letter we present the first co-aligned TRACE and the Solar Ultraviolet Measurement of Emitted Radiation (SUMER) observations of these loops. Although these loops appear static in the TRACE images, SUMER detects line-of-sight flows along the loops of up to 40 km s-1. The presence of flows could imply an asymmetric heating function; such a heating function would be expected for heating that is proportional to (often asymmetric) footpoint field strength. We compare a steady flow solution resulting from an asymmetric heating function to a static solution resulting from a uniform heating function in a hypothetical coronal loop. We find that the characteristics associated with the asymmetrically heated loop better compare with the characteristics of the loops observed in the TRACE data.