Internal DLA is a discrete model of a moving interface. On the cylinder graph ZN×Z, a particle starts uniformly on ZN×{0} and performs simple random walk on the cylinder until reaching an unoccupied site in ZN×Z≥0, which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order N2log⁡N particles. For a lower bound, we show that at least order N2 particles are required to forget which of two independent typical subsets the process started from.