High-order harmonic generation stands as a unique nonlinear optical up-conversion process, mediated by a laser-driven electron recollision mechanism, which has been shown to conserve energy, momentum, and spin and orbital angular momentum. Here we present theoretical simulations which demonstrate that this process also conserves a mixture of the latter, the torus-knot angular momentum $J_\gamma = L + \gamma S$, by producing high-order harmonics with driving pulses that are invariant under coordinated rotations. We demonstrate that the charge $J_\gamma$ of the emitted harmonics scales linearly with the harmonic order, and that this conservation law is imprinted onto the polarization distribution of the emitted spiral of attosecond pulses. We also demonstrate how the nonperturbative physics of high-order harmonic generation affect the torus-knot angular momentum of the harmonics, and we show that this configuration harnesses the spin selection rules to channel the full yield of each harmonic into a single mode of controllable orbital angular momentum.