Using a special version of the PCT-theorem which was found by Bisognano and Wichmann for finite-component Wightman fields, a proof of the spin-statistics theorem is given within the algebraic framework for quantum field theory. The proof covers massive bosons and fermions with ordinary as well as with parastatistics and, in contrast to earlier proofs, also works in 1+2 spacetime dimensions. Two uniqueness theorems concerning the Bisognano-Wichmann symmetries whose P_1_CT-part is used in the discussion of the spin-statistics theorem are presented for the algebraic setting. A derivation of the Bisognano-Wichmann result from standard assumptions of the algebraic setting is still lacking. The uniqueness theorems show that the operators which were found to implement the P_1_CT-symmetry and the Lorentz boosts in the Bisognano-Wichmann setting cannot implement any other symmetry on a local net of observables than precisely the one found by Bisognano and Wichmann. The analysis uses the notions of localization regions not only for algebras of observables, but also for single local observables. It is shown how these regions can be defined, and for the localization region of a single local observable it is investigated under what assumptions observables localized in spacelike separated region commute.