We present Submillimeter Array (SMA) mapping of 12CO J = 2 → 1, 13CO J = 2 → 1, and CN N = 2 → 1 emission from the ring-like planetary nebula NGC 3132, one of the subjects of JWST Early Release Observation near-infrared imaging. The ∼5″ resolution SMA data demonstrate that the Southern Ring’s main, bright, molecule-rich ring is indeed an expanding ring, as opposed to a limb-brightened shell, in terms of its intrinsic (physical) structure. This suggests that NGC 3132 is a bipolar nebula viewed more or less pole-on (inclination ∼15°–30°). The SMA data furthermore reveal that the nebula harbors a second expanding molecular ring that is aligned almost orthogonally to the main, bright molecular ring. We propose that this two-ring structure is the remnant of an ellipsoidal molecular envelope of ejecta that terminated the progenitor star’s asymptotic giant branch evolution and was subsequently disrupted by a series of misaligned fast, collimated outflows or jets resulting from interactions between the progenitor and one or more companions.