Multiwavelength observations of Type II supernovae have shown evidence of the interaction between supernovae and the dense slow winds from the red supergiant progenitor stars. Observations of planetary nebulae and the nebula around SN 1987A show that the slow winds from extended stars frequently have an axisymmetric structure with a high density in the equatorial plane. We have carried out numerical calculations of the interaction of a supernova with such an axisymmetric density distribution. For small values of the angular density gradient at the pole, the asymmetry in the interaction shell is greater than, but close to, that expected from purely radial motion. If the angular density gradient is above a moderate value, then the flow qualitatively changes and a protrusion emerges along the axis. For a power-law supernova density profile, the flow approaches a self-similar state in which the protrusion length is 2-4 times the radius of the main shell. The critical density gradient is larger for steeper density profiles of the ejecta. Most of our calculations are axisymmetric, but we have carried out a three-dimensional calculation to show that the protrusion is not a numerical artifact along the symmetry axis. For typical supernova parameters, the protrusions take ~> several years to develop. The appearance of the shell with protrusions is similar to that observed in VLBI radio images of the remnant 41.9 + 58 in M82 and possibly of SN 1986J. We also considered the possibility of asymmetric ejecta and found that it had a relatively small effect on the asymmetry of the interaction region.