WSimp[ℰ_]:=ℰ/.W[l___,i_,j_,r___]/;i+j==0:>WSimp@W[l,r]

WSimp[W[1,2,3,-3,-2,4]+W[1,-1,2,-2]]

Unprotect[NonCommutativeMultiply];
0**_=0;
_**0=0;
(a_+b_)**c_:=a**c+b**c;
a_**(b_+c_):=a**b+a**c;
w1_W**w2_W:=WSimp[Join[w1,w2]];
a_**W[]:=a;
W[]**a_:=a;
(a_*b_)**c_/;FreeQ[a,W]:=a(b**c);
a_**(b_*c_)/;FreeQ[b,W]:=b(a**c)

W[1,2,3]**W[-3,-2,4]

(OverBar[W[i_,is__]]):=(OverBar[W[i]])**W[is]+(OverBar[W[is]]);
(OverBar[W[i_]])/;i<0:=-(OverBar[W[-i]])**W[i]

(OverBar[W[1,2,3,-4]])

W[is___]**(OverBar[W[i_]]):=(OverBar[W[is,i]])-(OverBar[W[is]])

W[1,2,3]**(OverBar[W[2]])

(OverBar[W[2]])**(OverBar[W[3]])

(OverBar[W[2]])**(OverBar[W[3]])**W[1]

W[1]**(OverBar[W[2]])**(OverBar[W[3]])

(OverBar[W[1,2,-3]])**W[4]

W[1,2]**(OverBar[W[-1,2]])**(OverBar[W[2,-3]])

TPReduce[ρ_,ℰ_]:=ℰ//.{
(ws_**W[js___,j_])⊗u_/;j>0:>(ws**W[js])⊗(u/.ρ[(Subscript[σ, j,0])]),
(ws_**W[js___,j_])⊗u_/;j<0:>(ws**W[js])⊗(u/.ρ[(Subscript[(OverBar[σ]), -j,0])])
}