I recognise the piece called a 'Rook' here. The idea of a piece moving straight forward/backward/sideways on a hex board was put to me when I claimed that there was 'no piece bound to half a hex board' although the Dabbaba is bound to a quarter of it. I eventually documented it in my piece article Man and Beast 12: Alternative Fronts - you can find it by searching on that page for Moorhen. It is defined by the straightness of the directions (and of up and down on a hex-prism 3d board) regardless of whether they are orthogonal or hex diagonal and whether the orientation is Wellisch, Glinsky, or hex-ranked.
The labels for the directions are somewhat confusing as 'oblique' usually indicates a direction such as that of the FIDE Knight, going through intervening cells off-centredly. A more accurate description for the directions of each colour are forward/backward hex-diagonal, sideways orthogonal, forward/backward orthogonal, and sideways hex-diagonal. The linepieces in these directions I term Unicoranker, Rookfiler, Rookranker, and Unicofiler. These definitions also work on a Glinsky board, but on that it is the first two that have four directions and the second two only two.
The labels for the directions are somewhat confusing as 'oblique' usually indicates a direction such as that of the FIDE Knight, going through intervening cells off-centredly. A more accurate description for the directions of each colour are forward/backward hex-diagonal, sideways orthogonal, forward/backward orthogonal, and sideways hex-diagonal. The linepieces in these directions I term Unicoranker, Rookfiler, Rookranker, and Unicofiler. These definitions also work on a Glinsky board, but on that it is the first two that have four directions and the second two only two.