  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.
