The Pawns and King are standard, and do not see the board as the cylindrical board.
The Rooks are replaced by Waffles, but of course these are cylindrical Waffles; thus, from h1, an oWoA can go to h1 or to b3 in addition to its normal (non-cylindrical) destinations of f3, h2, and g1.
The Knights are cylindrical Knights, oN.
The Bishops are cylindrical Bishops, oB. Experienced players of Cylindrical Chess believe that the oB is worth almost as much as a normal (non-cylindrical) Rook.
The Queen is a cylindrical Chancellor, oRoN.
Because they have not been extensively playtested, you have the chance to be a pioneer....