This paper addresses the interaction of initially antiparallel vortex tubes containing an axial flow that induces a twisting of the vortex lines around the tube axes, using numerical simulations. Vortex tube configurations with both the same and opposite senses of twist - corresponding to the same and opposite signs of kinetic helicity density - are considered. It is found that the topology of the reconnection process is very different between the two cases. For tubes with the same sense of twist, the reconnection is fully three-dimensional (3D): vortex lines reconnect at a finite angle, and 3D vortex null points may be created. Following reconnection the vortex line topology in both bridge and thread structures exhibits a high degree of complexity. For oppositely twisted tubes the reconnection is locally two-dimensional, occurring along vorticity null lines, that in contrast to the untwisted case are not perpendicular to the tube axes. This leads to a break in the symmetry between the two vortex bridges generated during reconnection. For all cases studied, increasing the twist in the vortex tubes leads to a later, faster, and more complete reconnection process.