I'm curious about why the symbols O-O and O-O-O are used for castling in algebraic notation. Why not use the consistent and logical Kg1/Kc1 for White and Kg8/Kc8 for Black? Why adopt the less than obvious O-O and O-O-O from descriptive notation?
The castling notation was invented by Johann Allgaier and used for the first time in his 1811 2nd edition of his Neue theoretisch-praktische Anweisung zum Schachspiel.
He didn't explain why he came up with it.
Allgaier (and algebraic notation in general) used digit-0. The use of letter-O is an anglo-saxon oddity.
In my opinion, 0-0 and 0-0-0 are used to differentiate castling from ordinary king moves. The castling maneuver then stands out in the game notation, as opposed to say Kc8 which appears - on the surface anyway - to be an ordinary, one square, king move - until you look closer to see if the king is actually moving more than one square. It also helps to clearly indicate king-side (short) or queen-side (long) castling at a glance.
I think - at least as compared to things like ?! or !? it makes alot of sense. Oddly, its the only carryover from the Descriptive (P-K4) to Algebraic (e4) notation.
Kg1/Kg8 or Kc1/Kc8 are Singular moves where the King only moves during the Game . Since before the Algebraic Notation Descriptive Notation was used which was quite cumbersome & lengthy Algebraic was a short hand . Why 0-0 & 0-0-0 was exactly used is quite inexplicable but this is the only move in International Chess where two pieces are moving simultaneously . In K-side Castling K is moving two squares and in Q-side Castling it is moving three Squares .
So it might have seen an innovative way of recording the Castling move since King is the most important piece in a Chess Game 0-0/0-0-0 was a royal move indicating King is going inside the castle and the Battle Begins .