To contribute (hopefully clarifying, but at risk of doing the opposite) to the discussion of the pieces:
It is established that a triangular‐celled board is isomorphic to a hex‐cell board with one hex‐‘bishop’ binding inaccessible. On such a board, the usual triangular ‘rook’ — the same rook present here — becomes a crooked rook making 60° turns between steps (known to Gilman as a Longgirlhexer). The usual colourbound piece, moving in a straight line on triangles of the same colour to every second ‘rook’ cell is then equivalent to the hex ‘bishop’, while the usual ‘third’ rider, moving in a straight line alternating steps through the sides and across the corners, is a normal hex rook that can jump over the gaps (or, equivalently, a straight wazir–dabbaba alternator which I haven't seen named)
This game's bishop analogue is then equivalent to the crooked hex ‘bishop’ (Gilman's Longrangehexer), hence bearing the same relation to the rook analogue as the usual hex ‘bishop’ to the hex rook
The knight analogue here is equivalent to a hex dabbaba + hex ‘knight’; the pawn analogue is more unusual
To contribute (hopefully clarifying, but at risk of doing the opposite) to the discussion of the pieces:
It is established that a triangular‐celled board is isomorphic to a hex‐cell board with one hex‐‘bishop’ binding inaccessible. On such a board, the usual triangular ‘rook’ — the same rook present here — becomes a crooked rook making 60° turns between steps (known to Gilman as a Longgirlhexer). The usual colourbound piece, moving in a straight line on triangles of the same colour to every second ‘rook’ cell is then equivalent to the hex ‘bishop’, while the usual ‘third’ rider, moving in a straight line alternating steps through the sides and across the corners, is a normal hex rook that can jump over the gaps (or, equivalently, a straight wazir–dabbaba alternator which I haven't seen named)
This game's bishop analogue is then equivalent to the crooked hex ‘bishop’ (Gilman's Longrangehexer), hence bearing the same relation to the rook analogue as the usual hex ‘bishop’ to the hex rook
The knight analogue here is equivalent to a hex dabbaba + hex ‘knight’; the pawn analogue is more unusual