Pin groups in physics

The double covers $\mathrm{Pin}\mathrm{}(s,t)\mathrm{}$ and $\mathrm{Pin}\mathrm{}(t,s)\mathrm{}$ of the orthonormal groups $\mathrm{O}\mathrm{}(s,t)\mathrm{}$ and $\mathrm{O}\mathrm{}(t,s)\mathrm{}$, respectively, are not necessarily isomorphic. They are locally isomorphic but, in general, not globally so. We compute the vacuum expectation values of several quantized fermionic currents on a nonorientable spacetime ($\mathrm{Klein}\mathrm{bottle}\times R^2$) for both groups. They are strikingly different.