We classify two-qudit Clifford dual-unitary gates over the finite field F_q, where the local dimension q is a prime power, and apply the classification to exact local masking and operator transport in homogeneous brickwork circuits. Under ordered one-qudit Clifford equivalence, the dual-unitary locus contains q-2 perfect-tensor cores, one rank-one core, and one SWAP core. Homogeneous repetition separates these cores into five distinct transport phases. The one-site Weyl edge channels determine exact local-masking distances. Writing dᵣ(t) for the masking distance against output observers controlling at most r sites, perfect-tensor circuits attain d₁(t)=4t, d₂(t)=4t-2, whereas delayed erasers satisfy d₁(t)=4t-2, d₂(t)=4t-4 for t≥ 2. Consequently, sufficiently short quantum messages are completely hidden from every one- or two-qudit output subsystem, even when the input is entangled with a reference, while remaining exactly recoverable from the full output. For q=3, we construct an explicit perfect-tensor Clifford gate from two inverse SUM gates. Exhaustive Weyl-support searches for t=1,2,3 reproduce the predicted masking distances. For a coherent perturbation of this gate, local leakage scales linearly with the perturbation strength, whereas the infidelity of recovery using the ideal inverse scales quadratically near the perfect point.