High-resolution HST imaging of the compact planetary nebula NGC 6644 has revealed two pairs of bipolar lobes and a central ring lying close to the plane of the sky. From mid-infrared imaging obtained with the Gemini Telescope, we have found a dust torus which is oriented nearly perpendicular to one pair of the lobes. We suggest that NGC 6644 is a multipolar nebula and have constructed a 3-D model which allows the visualization of the object from different lines of sight. These results suggest that NGC 6644 may have similar intrinsic structures as other multipolar nebulae and the phenomenon of multipolar nebulosity may be more common than previously believed.