In this work, we introduce a theory of stochastic integration with respect to symmetric α-stable cylindrical Lévy processes. Since α-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This approach enables us to characterise the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to an α-stable cylindrical Lévy process as the collection of all predictable processes with paths in the Bochner space L^α. We demonstrate the power and robustness of the developed theory by establishing a dominated convergence result allowing the interchange of the stochastic integral and limit.