"What I meant is that a4' cannot be reached from a3."
Why not? A rook, pawn or queen starting from a3 and occupying the switch must decide which square of the switch is to occupy - 4 or a4.
"a move into the lower board half would always look as if you started from a4."
That's not true: A rook on a8 that wants to move to the lower half of the board can occupy square 4, but not a4. A rook on b8 can occupy a4, but not 4. Incidentally, in this direction the distinction between 4/a4 makes no sense for a rook (or queen). The switch only works in one direction. Either from the lower half of the board starting from a1, a2, a3, or from the upper half of the board starting from h8, h7, h6.
"What I meant is that a4' cannot be reached from a3."
Why not? A rook, pawn or queen starting from a3 and occupying the switch must decide which square of the switch is to occupy - 4 or a4.
"a move into the lower board half would always look as if you started from a4."
That's not true: A rook on a8 that wants to move to the lower half of the board can occupy square 4, but not a4. A rook on b8 can occupy a4, but not 4. Incidentally, in this direction the distinction between 4/a4 makes no sense for a rook (or queen). The switch only works in one direction. Either from the lower half of the board starting from a1, a2, a3, or from the upper half of the board starting from h8, h7, h6.