I have heard it explained that, within chess, the movement of the pieces represents how their loyalty is split between the Church and the Crown. Orthogonal movement represents political (Crown) affiliation, and diagonal movement represents religious (Church) affiliation.
So, Bishops are exclusively loyal to the Church, while Rooks are exclusively loyal to the Crown. The King and Queen serve both, but not at the same time. Pawns move at the command of the Crown, but fight for their Church. Knights, with their code of chivalry, are sworn to serve both equally. So the Knights' L-Shaped movement is actually a political action combined with a religious action.
It may be utter hokum, but it makes a surprisingly good explanation.
Bearing that in mind, I propose the following new pieces:
the nihilist. Doesn't move in any direction, diagonal or straight.
the swinging voter. Moves straight along the north-south axis on its first move. Must move along the east-west axis on its next move. Then N-S, then E-W, etc...
the John Howard. Can move only backwards. In addition, the white one must refuse to be sorry whenever it takes a black piece.
the Peter Garrett. A pawn with a shiny dome. Can move only one space at a time but is never sure of the direction.
the Nicole Cornes. This piece is not permitted to take any other pieces. As such it is used only as part of a sacrifice.
the crusader. Is able to move diagonally as far as the edge of the board, taking all pieces in its path, regardless of colour.
the fundamentalist suicide bomber. moves diagonally forward one square at a time. May instead "explode", taking out all pieces within a two-square radius, including itself.