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Solid molecular hydrogen has been predicted to be metallic and high-temperature superconducting at ultrahigh hydrostatic pressures that push current experimental limits. Meanwhile, little is known about the influence of nonhydrostatic conditions on its electronic properties at extreme pressures where anisotropic stresses are inevitably present and may also be intentionally introduced. Here we show by first-principles calculations that solid molecular hydrogen compressed to multimegabar pressures can sustain large anisotropic compressive or shear stresses that, in turn, cause major crystal symmetry reduction and charge redistribution that accelerate bandgap closure and promote superconductivity relative to pure hydrostatic compression. Our findings highlight a hitherto largely unexplored mechanism for creating superconducting dense hydrogen, with implications for exploring similar phenomena in hydrogen-rich compounds and other molecular crystals.