Quantum indistinguishability: alternative constructions of the transported basis

Our earlier arguments (Berry M V and Robbins J M 1997 Proc. R. Soc. Lond. A 453 1771-90) leading to the spin-statistics relation are summarized and then revisited. Constructions are described that satisfy all our previous requirements but lead to the wrong exchange sign (one such alternative is the replacement of commutators by anticommutators in the Schwinger representation of the spins of the particles); we suggest why these might be unacceptable.