Not only spinors can be used to demonstrate that $360^\circ$ is not an identity transformation. See http://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di991816.99p0601v.pdf (The Mercedes Knot Problem, by Aleksandar Jurisic) and http://www.pha.jhu.edu/~kgrizz/Documents/10-2-13/spinorSpanner.pdf (The Spinor Spanner, by Ethan D. Bolker).