Deriving the equations for two-electron integrals in Hartree-Fock theory
I believe your statement


  I can permute the electrons to achieve the same form, but that would introduce a negative sign.


is mistaken. The exchange integrals arise from permuting one pair of integration variable instances, not two. The latter is the case necessary here. Szabo and Ostlund state on the same page:


  Since $r_{12} = r_{21}$, we can interchange the dummy variables of integration in the second term [...].


where the second term appears to be you are troubled with.