Oh, I think I get it.
With a 45° slice ◤, you mirror on X 🭭 then on Y 🮚 and then you rotate 90° (integer matrix) 🮛.
With a 22.5° slice, you mirror on X then on Y and then you rotate 90° ✠ and you have to rotate 45° (√2) (or 22.5° (√(2±√2)) at the start).
With a 45° slice ◤, you mirror on X 🭭 then on Y 🮚 and then you rotate 90° (integer matrix) 🮛.
With a 22.5° slice, you mirror on X then on Y and then you rotate 90° ✠ and you have to rotate 45° (√2) (or 22.5° (√(2±√2)) at the start).