The software we use to generate the "flower petal" projection is an in-house software package called ISIS2. This software is currently only supported on Linux (Suse9 - works with Red Hat) and Sun Solaris (OSX soon). The source code for ISIS2 is free can be used by all. The actual program is called "daisy".
By following the directions below one could use other cartographic software to emulate the daisy program. (ie. make all the pieces and combine them in photoshop or other image package). One free software package that can project raster to an from different projections in windows is Generic Mapping Tools (GMT). This software is command line driven and can be confusing but it is very powerful. Any commercial software package that supports image reprojection and the two projection listed below could also be used.
Link to GMT: http://gmt.soest.hawaii.edu/
Link to ISIS2 (and ISIS3): http://astrogeology.usgs.gov/Projects/ISIS/
Here is the ISIS help on Daisy:
HELP INFORMATION - proc "daisy" library "$ISISEXE"
DAISY - Create a "flower petal" projection for globes
Create a series of 12 Transverse Mercator projections that are 32 degrees in longitude and 91 degrees in latitude to form a "flower petal" projection for making globes. If the northern hemisphere is being created, the latitude range is -1 to +90. If the southern hemisphere is being created, the latitude range is -90 to 1. The petals are formed by changing the center longitude and map rotation in steps of 30 degrees for each Transverse Mercator projection. Each section is put into the mosaic. The final center of the projection is a Lambert Azimuthal Equal-area centered on the pole and covers from -90 to -75 or 75 to 90 degrees latitude and 360 degrees in longitude. This is the last cube that is placed in the mosaic.
The programs that are used are: NUPROJ, GEOM, MOSAIC and DAISPARM (a special program that calculates parameters for DAISY).