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[Edited to correct the final position (which didn't work before: 2 Bh1+! Bg2 3 Rf1 Bb1 etc.), and — while I'm at it — to remove a few superfluous pawns]

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,d3,f3,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain not ended by stalemate, remove the White and Black bishops on g2 and h1, put the White king on h1, and add Black pawns on h2 and g3:

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2

and now 4 Kg2 Bf1+ 5 Kh1 Be2 etc. This extends naturally to wider boards, with a correspondingly longer sequence between 1 Ra1 and the onset of forced repetition.

[Edited to correct the final position (which didn't work before: 2 Bh1+! Bg2 3 Rf1 Bb1 etc.), and — while I'm at it — to remove a few superfluous pawns]

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,d3,f3,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

_{(source: janko.at)}

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain not ended by stalemate, remove the White and Black bishops on g2 and h1, put the White king on h1, and add Black pawns on h2 and g3:

_{(source: janko.at)}

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2

_{(source: janko.at)}

and now 4 Kg2 Bf1+ 5 Kh1 Be2 etc. This extends naturally to wider boards, with a correspondingly longer sequence between 1 Ra1 and the onset of forced repetition.

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,c3,d3,...,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain, remove the Black Bh1 and add White Rf2 and Black Kh2 and Ng1:

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2 (not Ne2 stalemate)

and now 4 Bf1+ Kh1 5 Bg2+ Kh2 etc. (4 Rf1 or 5 Rg2 would stalemate, and 5 Rh2 would be checkmate in this chess variant).

[Edited to correct the final position (which didn't work before: 2 Bh1+! Bg2 3 Rf1 Bb1 etc.), and — while I'm at it — to remove a few superfluous pawns]

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,d3,f3,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain not ended by stalemate, remove the White and Black bishops on g2 and h1, put the White king on h1, and add Black pawns on h2 and g3:

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2

and now 4 Kg2 Bf1+ 5 Kh1 Be2 etc. This extends naturally to wider boards, with a correspondingly longer sequence between 1 Ra1 and the onset of forced repetition.

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,c3,d3,...,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain, remove the Black Bh1 and add White Black Kh2 and Ng1:

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2 (not Ne2 stalemate)

and now 4 Bf1+ Kh1 5 Bg2+ Kh2 etc. (4 Rf1 or 5 Rg2 would stalemate, and 5 Rh2 would be checkmate in this chess variant).

Here's yet another mechanism, which also allows arbitrarily long chains of irreversibility. The new construction is suggested by what may be an unintenional feature/bug in the original question:

[...] the game must be extendable at least two moves in both direction from both A and B.

The number seems arbitrary (why two moves rather than three or 23?), and I'm guessing that the intention was to guarantee that one can continue indefinitely forwards or backwards from both positions but give an easily checked criterion. However, it's possible for a position to be extendable $n$ moves and no further for any given $n$, as long as we're allowed an arbitrarily large board and supply of pieces (as the question proposal did explicitly allow). For example: let Position A have White rook a5, bishops c2,e2,g2, and pawns b3,c3,d3,...,h3 vs. Black bishops b1,d1,f1,h1, plus any immobile setup that traps both kings above the 3rd row:

and play 1 Ra1 to get Position B. Not only can't we get back from B to A, but after a few moves (but more than 2) we must reach stalemate: 1...Ba2 2 Bb1 (or 2 Rb1 stalemate, or 2 Rc1 Bc2 stalemate) 2...Bc2 3 Bd1 Be2 4 Bf1 and now 4...Bg2 is stalemate.

For a long irreversibility chain, remove the Black Bh1 and add White Rf2 and Black Kh2 and Ng1:

Now after 1 Ra1 play can continue indefinitely, but only with 1...Ba2 2 Bb1 Bc2 3 Bd1 Be2 (not Ne2 stalemate)

and now 4 Bf1+ Kh1 5 Bg2+ Kh2 etc. (4 Rf1 or 5 Rg2 would stalemate, and 5 Rh2 would be checkmate in this chess variant).