NN = 200; R = 0.05;
xlist = Table[x, {x, -0.2 [Pi], 0.2 [Pi], 0.01}];
modl[n_] := 2*^-3 (Quotient[n, 2] - NN/2);
t1 = -1 + Cos[x] - I Sin[x] + I R; t1p = -1 + Cos[x] + I Sin[x] + I R;
t2a[n_] := -1 - modl[n]; t2b[n_] := -1 + modl[n];
mat[x_] =
  DiagonalMatrix[
    Table[If[EvenQ[n], t1, t2a[n]], {n, 0, 2 NN - 1 - 1}], 1] +
   DiagonalMatrix[
    Table[If[EvenQ[n], t1p, t2a[n]], {n, 0, 2 NN - 1 - 1}], -1] +
   DiagonalMatrix[
    Table[If[EvenQ[n], t2b[n], 0], {n, 0, 2 NN - 1 - 3}], 3] +
   DiagonalMatrix[
    Table[If[EvenQ[n], t2b[n], 0], {n, 0, 2 NN - 1 - 3}], -3];
list0 = Sort@Re@Eigenvalues[mat[xlist[[3]]]];
list0p = Table[list0[[i]] + list0[[2 NN - i + 1]], {i, NN}];
ListPlot[Tooltip@list0p, PlotRange -> All]