There exist many remarkable pieces that function finely in a
context of Western pieces. The below links go to Game Courier
presets. The movement rules are wholly enforced, which was
necessary as it would otherwise be too difficult to make use of
these newfangled pieces. In order to learn how they work it's a
good idea to move the pieces about on the following preset
boards, and see what happens (press 'Move'). To see the rules,
press 'Rules'.

The Swedish Cannon is essentially a Korean Cannon while it both
captures and moves by jumping. However, if the Korean Cannon has
no jump available, then it becomes immobile. The Swedish Cannon,
however, acquires the right to move as a rook (but not capture).
As soon as there is a jump available, it loses this right. This
makes it possible to introduce an interesting orthogonal cannon in
a Western context. Its value corresponds probably to a bishop.
Gustavian Cannon Chess

Another Cannon, easier to comprehend, is the Culverin. The
Culverin can step one square in any direction like a king,
provided that the square is empty. It can capture long-distance
by leaping over any piece in any direction. Enemy pawns, however,
can restrict its movement. The Culverin can only reach the
square immediately behind the enemy pawn, regardless of
direction. By example, if the black king is checked on the
g-file, it can defend itself by moving the pawn forward. While
its tactical capacity is truncated the Culverin can harmonize
with the positional qualities of the game. In Culverin Chess the
external cannon must, in any move, be dropped on an empty square
on the first rank immediately behind a friendly pawn, which then
automatically moves forward one step. This mildly enforces
introduction. Its value corresponds, likely, to a light piece.
Culverin Chess

The Zeppelin is an orthogonal version of Schmittberger's
Airplane, but perhaps more useful. It can fly immediately
to any empty square orthogonally. This means that it can also fly
directly, from one of the extra corner squares, to the opposite
corner square in the enemy position. This is a move that one
should watch out for. Therefore extended castling is employed.
By flying over an enemy piece, and landing on an empty square
immediately behind it, the Zeppelin makes a capture.
Zeppelin Chess

The Adjutant is essentially an enhanced bishop while it can only
move on the same square colour. Despite appearances it is a
sliding piece, not a jumping piece. It slides on the diagonal and
orthogonal squares of the same colour. On the orthogonals it
simply ignores the other square colour. Of course, this creates a
very mobile piece, possibly worth as much as a rook.
Adjutant Chess

The Divaricator is an odd one. It can slide two steps in any
direction. But it captures by oblique withdrawal. It must
withdraw diagonally to capture orthogonally, vice versa. Thus it
has twice as many capture directions as a standard withdrawal
piece, which makes it rather strong, possibly four pawns on a
standard board.
Accessory Chess

The Turret is a catapult, capable of slinging any piece
immediately behind itself to a square before the Turret's
destination square. A pawn can be slung to the promotion square,
automatically promoting to queen. A pawn cannot be flung to the
first rank. A rook slung from its initial square and back, does
not lose its castling right. This piece, which moves as a queen,
and can only capture by catapulting, sounds like sheer madness,
but it actually works. It is perhaps worth 4, but I really don't know.
Try it here:
Turret Chess

The Oxybeles is yet another catapult, but this one is lethally
dangerous as it can sling pieces all across the board, a pawn
too, which immediately promotes. Its weakness is its slow
movement. It can only move one step at a time, like a king. A
pawn cannot be slung to the first rank. This piece, which can
only capture by catapulting, has interesting tactical features. A
piece, located immediately behind the catapult, can be slung to
an attractive position. Neither the Turret nor the Oxybeles can
capture a friendly piece by catapulting. So there is never any
danger in moving toward a friendly piece. Any piece catapulted by
the Oxybeles, will land before the friendly piece.
Try it here:
Oxybeles Chess

Finally, I have implemented one of my bifurcation pieces.
The Secutor slides on the orthogonals like a rook. It captures by
colliding against any piece and then deviating to any of
the two adjacent diagonals (in the prolonged movement direction).
Thus to capture, the Secutor jumps directly to an enemy piece
and lands on it, provided that any intermediate squares are empty.
The Secutor's value is 3 (my estimate). This piece opens up new
tactical vistas. Its moves are wholly enforced, which is necessary
as it moves in two stretches.
Secutor Chess
/Mats