The quantum SWITCH is an example of a process with an indefinite causal structure, and has attracted attention for its ability to outperform causally ordered computations within the quantum circuit model. To date, realisations of the quantum SWITCH have relied on optical interferometers susceptible to minute path length fluctuations, complicating their design, limiting their performance and posing an obstacle to extending the quantum SWITCH to multiple parties. In this Letter we overcome these limitations by demonstrating an intrinsically stable quantum SWITCH utilizing a common-path geometry facilitated by a novel reciprocal and universal $\mathrm{SU}(2)$ polarization gadget. We certify our design by successfully performing a channel discrimination task with near unity success probability.