Tubular auxetic structures have wide-ranging applications including medical stents, collapsible energy absorbers, and novel fasteners. To expand development in these areas, and open up new application directions, an expanded range of design and construction methods for auxetic tubes is required. In this study we propose a new method to construct polygonal cross-section auxetic tubes using the principles of origami and kirigami. These tubes exhibit useful global auxetic behaviour under axial extension, despite the individual polygon faces not being auxetic themselves. A flat kirigami sheet cannot be simply folded into a polygonal tube since this creates kinematic incompatibilities along the polygon edges. This is resolved by replacing the edge folds with an origami mechanism consisting of a pair of triangular facets. This approach eliminates the incompatibilities at the edges while maintaining a connection between faces. The proposed edge connection also introduces additional control parameters for the tube kinematics: for example, introducing a kinematic limit on tube extension and enabling non-uniform behaviour along the length of the tube. The rich kinematic behaviour possible with polygonal cross-section kirigami tubes has potential applications ranging from soft robotics to energy-dissipating devices.