  In this work, we introduce a theory of stochastic integration with respect to
symmetric $\alpha$-stable cylindrical L\'evy processes. Since $\alpha$-stable
cylindrical L\'evy 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 $\alpha$-stable cylindrical L\'evy process as the
collection of all predictable processes with paths in the Bochner space
$L^\alpha$. 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.
