We present James Webb Space Telescope (JWST) Near-Infrared Camera (NIRCam) Brα, H2, [Fe II], and polycyclic aromatic hydrocarbon (PAH) imaging of the molecule-rich, high-excitation bipolar planetary nebula (PN) NGC 6537 (the Red Spider), complemented by new Atacama Large Millimeter/submillimeter Array (ALMA) and Chandra X-ray Observatory (Chandra) observations and archival Hubble Space Telescope (HST) images. The resulting multiwavelength view of the Red Spider establishes the detailed lobe─torus structure of the nebula and the mass-loss history of its progenitor star. The extinction-penetrating JWST/NIRCam Brα and PAH and ALMA 3 mm continuum imaging exposes the complexity of the ionized inner nebula. JWST/NIRCam H2 imaging traces the full, ∼1.1 pc extent of the bubble-like lobes formed by fast (∼300─400 km s−1) polar outflows, while ALMA 13CO(1─0) mapping reveals a point-symmetric, slowly (∼10 km s−1) expanding equatorial torus of radius ∼ 0.13 pc. In striking contrast, the [Fe II] image displays an extended S-shaped emission morphology that traces collisions between an active, collimated wind and slower-moving material along the lobe rims. No X-rays are detected from the nebula or its central star in deep Chandra/HRC-I imaging. However, the combined HST and JWST imaging reveals a near-IR excess at the central star indicative of emission from hot (∼1000 K) circumstellar dust. We propose that interactions between the nebular progenitor star and a close companion are responsible for the ejection of NGC 6537's molecular torus, the formation of a circumbinary dust disk, and the launching of fast, wandering, collimated outflows that have inflated the polar lobe bubbles traced by near-IR H2 emission and are presently generating the [Fe II]-emitting shocks.