The piece you call 'Squire' is commonly called 'Camel'. It featured under that name in a game called Tamerlane Chess, which was played in Persia (on an 11x10 board with two extra squares) in the 14th century.
The relation you noticed between R/N on the one hand and B/C on the other hand is known as 'conjugation'. The set of (say) dark squares on a checkered board has the same topology as the original board, after a 45-degree rotation. A Bishop move on the real board looks like a Rook move on the conjugate board, and similarly, a Camel move on the real board is a Knight jump on the conjugate board. One could checker the conjugate board again, to make a 'second-order' conjugate board out of half its squares. Moving like a Rook there would mean moving like a Rook, but skipping every odd square (jumping over whatever mgight be there). Such a piece would be severely color bound, with only access to 1/4 of the squares.
Note that in Chinese Chess a Horse is a Knight that really moves one square straight, followed by one diagonal: it can be blocked on the squares orthogonally next to it, even though it cannot move there. But not on those diagonally next to it, or two steps orthogonally away from it.
What you call Anti-Pawn is also known as 'Berolina Pawn'. En-passant capture is complex with Berolina Pawns. It can happen that you can make a non-capture Pawn move to a square that was just stepped over in a diagonal double-push of an opponent Anti-Pawn (which is impossible with FIDE Pawns). You would have to specify if that counts as a capture or not. Things are even more messy when you mix Pawns and Anti-Pawn, which is why Spartan Chess (where the Persians have FIDE Pawns, but the Spartans have Anti-Pawns) has no e.p. capture.
Somehow I missed this sub-forum the other day and tried to post this somewhere that it didn't belong.
I want to limit myself to one observation--I once saw a table of every concievable power of moving that a piece could posses--as for instance: "Rook+Knight" or "King + Knight" or whatever.
It seems to me that one very basic move (logically) is always ommited.
I've heard it said that a Knight move is a combination of moving one Square like a Rook and then one Square like a Bishop.
I never pictured it that way. I always thought that a Knight moved three Rook Squares except one Square was at Right Angles to the other two:
"One; Two; OVER" or "OVER; One; Two".
In other words, every Knight move is a "Broken" three square Rook move.
Imagine a Piece--call him a "Squire"--who moves two Squares Diagonally like a Bishop, and then moves one Square Diagonally at Right Angles to the first two moves.
If a Knight is a "Broken" Rook then a Squire is a "Broken" Bishop.
The Squire could cover distance a bit quicker than a Knight--but unlike a Knight, he only has access to half the board. If Knight and Bishop both rate "3 Points" a Squire is probably worth about "2 Points" or a bit more.
If you use a 10x10 board to make room for the Squires...Lets move the Pawn ranks forward one rank.
It seems obvious and I've heard Chess Variant enthusiasts say that as the number of Squares go up, you need to add a few more pieces lest the board have an empty feel.
Lets add another Squire (for a Total of Three) and a third and perhaps a fourth Knight in the rank between the pieces and pawns...
Ever hear of "Logical Chess" wherein Black's King and Queen are transposed so that Black's starting position is a mirror image of White's?
Most folks find it "Drawish". Somehow Chess' Assymetry contributes to Sharp and Interesting Play.
So IF Assymetry is good--If White's third Squire was on Black--for instance--then Black's Extra Squire would be on White...
Each side would be somewhat stronger on one Diagonal.
How about even More Assymetry?
I had started with the assumption that the Squires would probably go in between Knights and Rooks--since their Diagonal movement lets them cover the distance to the center faster than a Knight could.
But IF Assymetry is good, why not put both First Rank Squires Between the Knight and the Rook on the Queen Side? Once again, that Increases Assymetry and as a side benefit, it no longer take an extra move to Castle King Side. (Though Queen Side Castling becomes all but Impossible. One might try allowing Castling Queen Side as though one (or both) Back Rank Squires weren't there...That might or might not turn out to work well in actual play.)
I assume that the third--the solitary Second Rank Squire--would be somewhere on the King Side.
If we place a third and fourth Knight in the Knight Columns in the Second Rank it would allow many of the traditional Pawn Attacks on King and Queen Four and Five, reinforced by the same sort of Knight moves.
One more idea--Some Chess Variants have a sort of "Anti-Pawn"--He moves forward one Square Diagonally right or left, but he only captures straight ahead.
Whichever side the Second Rank Squire goes on, why don't we balance him with two or three "Anti-Pawns" on the other wing?
HMMMmmnn...?
Maybe "Anti-Pawns" that reach the Tenth Rank can Promote--but only to Squire...
{With the "Anti-Pawn's" choice of two possible squares to move to, he'd be harder to block than a Pawn.}
The only way to see if any of these ideas result in lively play would be to round up some folks willing to try them--but what does anyone think?
As an Aside:
Anyone ever play Raumschach?
I never did, but it looks interesting...
Saxon Violence