The setup and mate in four immediately reminded me of
the Babson Task
Since Black is threatening
1... Qxh2+ which at least delays mate by a couple of moves,
White's first move must be
Black now has
some moves with the knight and king, which I haven't analyzed but assume they'll lead to unthematic mates.
The problem is really about
the different promotions by the black e-pawn:
1... e1Q 2. axb8Q threatening all kinds of mate-in-twos, e.g. a discovered check from the queen on f8, or 3. Na6 Kxd5 4. Be4.
(A) 2... Qxe7 3. d7 Qd6 4. Qfxd6#
(B) 2... Qe5 3. Qa7+ Kxd6 4. fxe5#
(C) 2... Qxc3 3. Re4 Qxc1 4. d7#
1... e1R. Now 2. axb8Q Rxe7! delays the mate long enough, since 3. d7 (from line A) is now stalemate.
Promotion to rook doesn't help; 2. axb8R Re5 and Qa7 from line B is not available.
But promotion to bishop helps: 2. axb8B! Re5 3. Ba7+!
1... e1B. Now 2. axb8Q Bxc3 3. Re4 (line C) is stalemate, as it would be when White promotes to rook or bishop on move 2.
However, 2. axb8N works: bishop moves are met by 3. N8a6+ and 2... Kb6 by 3. Nd7+ Ka5 4. Qa8# (or 3... Ka7 4. Ra1#).
1... e1N. Now 2. axb8Q fails because 2... Nxd3 3. Bxd3 is stalemate.
By promoting to a rook, 2. axb8R Nxd3 3. Bxd3 Black can take on d6: 3... Kxd6 4. Rf7#.
So this is related to what Tim Krabbé calls a
cyclic Babson (also referred to in the title of this puzzle)
but not quite; here the promotions to queen do match each other. So:
if Black promotes to a queen, White promotes to a queen;
if Black promotes to a rook, White promotes to a bishop;
if Black promotes to a bishop, White promotes to a knight;
if Black promotes to a knight, White promotes to a rook.
A wonderful puzzle! (Even though capturing a piece on the key move, a queen no less, is usually considered bad form.)