The interplay between twist-angle engineering and unconventional magnetism provides a powerful new route to control quantum phenomena. We theoretically investigate a heterostructure comprising a d-wave superconductor proximitized by a two-dimensional d-wave altermagnet. We reveal that the momentum-space mismatch between the superconducting gap nodes and the altermagnetic spin-splitting nodes generates a robust, twist-tunable odd-frequency spin-triplet pairing. Consequently, the macroscopic electromagnetic response of the system can be tuned from a conventional diamagnetic Meissner state to an anomalous paramagnetic Meissner effect driven entirely by the interfacial twist angle. For a d_(x²-y²) altermagnet, the paramagnetic response is maximized at perfect alignment (φ=0) and completely suppressed at a maximal twist of φ=π/4, while a d_xy altermagnet exhibits the exact complementary behavior. Our results establish twisted altermagnetic heterostructures as a versatile platform for engineering odd-frequency pairing and macroscopic superconducting phases.