Optimal physical approximations of pure-state cloning and transposition are complementary channels
State cloning and state transposition are fundamental transformations that, despite being desirable, cannot be perfectly realised as a consequence of two distinct requirements of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, in spite of these disparate conceptual restrictions, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. This complementary relation allows us to determine the exact quantum and private classical capacities of the universal symmetric $N\to N+K$ cloning channel in the fully general multicopy qudit setting, thereby closing the remaining gap in the quantum capacity problem for universal cloning. We provide an explicit circuit that simultaneously implements optimal pure-state $N\to K$ transposition and optimal $N\to N+K$ symmetric cloning, both attaining the same fidelity. Additionally, we prove that the optimal $N\to K$ pure-state transposition channel is unique and realised by an estimation strategy. We also obtain the optimal mixed-state $N\to 1$ transposition, and discuss its differences from its pure-state analogue.