Altermagnetism establishes momentum-dependent spin splitting through non-symmorphic crystal symmetries, yet whether these same symmetries simultaneously govern spin and lattice collective excitations remains open. Here we show, using first-principles calculations and linear spin-wave theory, that monolayer CuF₂ hosts both chirality-split magnons and cycloidal phonons controlled by the same P2₁/c symmetry operations. The altermagnetic order drives strongly anisotropic magnon chirality via symmetric anisotropic exchange, with Dzyaloshinskii–Moriya interactions acting as a weak secondary modulation. Crucially, the phonon and magnon chiral responses are directionally complementary: cycloidal phonon angular momentum emerges precisely where magnon chirality is symmetry-suppressed, and vice versa. The magnon bands further carry quantized Chern numbers C^M = ± 2, confirming non-trivial altermagnetic topology. These results establish monolayer CuF₂ as a platform where a single symmetry framework engineers magnonic, phononic, and topological responses, providing a direct connection between altermagnetism and spin-lattice chirality in two-dimensional materials.