This combines three of my least favorite things in dataviz (circular encoding, rainbow coloring, and means out of context), so of course I have to try it out!
The simplest thing to do is probably to use the computed x/y points and make a scatter plot of colored dots around a circle.
A fancy solution is to make a custom shape file where each shape is a wedge of the circle. Sometimes people do that for wafer maps so that each component can have a color.
For the circle, you have to know how many wedges you want in advance and refer to each by name. Here's what that can look like (I couldn't bring myself to use the rainbow):
Attached is a script that makes three data tables for this chart. Two data tables comprise the shape descriptions and the third one has the data, most random in this case. As long as you're happy with the wedges, you can use those files for different data tables. The trick is to set a column property called Shape Use in your data table to point to the shape tables.
Having gone wild and making a polar chart, I must at least point out this paper, Graphical Inference for Infovis, by Hadley Wickham et al, which has an incidental result where Cartesian coordinates were more effect than polar coordinates for wind data. Admittedly, they were unwilling to simplify down to a mean and were trying to show all the data.