(A) reflection sigma -> -sigma mod m on types
   n=3: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=1: True
   n=4: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=2: True
   n=5: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=2: True
   n=6: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=3: True
   n=7: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=3: True
   n=8: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=4: True
   n=9: involution, maps winding k onto n-k, commutes with facets: True; fixed types in the middle component: 0; PGL components floor(n/2)=4: True
(B) GF(2) cohomology of Delta'_{n/2} and of its quotient by the reflection (H^i = H^BM_{n-1-i})
   n=4: 8 cells, 4 orbits; H^*(Delta') = {0: 1, 1: 1} [S^1: True]; H^*(Delta'/refl) = {0: 1, 1: 1} [RP^1: True]; 0.0s
   n=6: 300 cells, 150 orbits; H^*(Delta') = {0: 1, 3: 1} [S^3: True]; H^*(Delta'/refl) = {0: 1, 1: 1, 2: 1, 3: 1} [RP^3: True]; 0.0s
   n=8: 22792 cells, 11396 orbits; H^*(Delta') = {0: 1, 5: 1} [S^5: True]; H^*(Delta'/refl) = {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1} [RP^5: True]; 0.4s
(C) GF(2) cohomology of E/G from the E-type cells (H^i = H^BM_{n/2-i})
   n=4: 2 E-type cells; H^*(E/G;GF(2)) = {0: 2} [S^0: True]
   n=6: 12 E-type cells; H^*(E/G;GF(2)) = {0: 1, 1: 1} [S^1: True]
   n=8: 74 E-type cells; H^*(E/G;GF(2)) = {0: 1, 2: 1} [S^2: True]
   n=10: 540 E-type cells; H^*(E/G;GF(2)) = {0: 1, 3: 1} [S^3: True]
   n=12: 4682 E-type cells; H^*(E/G;GF(2)) = {0: 1, 4: 1} [S^4: True]
   n=14: 47292 E-type cells; H^*(E/G;GF(2)) = {0: 1, 5: 1} [S^5: True]
