You probably already found this since your post was so long ago, but is your Ginger Lily Hedychium gardneranum? I see Ginger Lilies listed as both Hedychium and Alpinia, and both those genuses (geni??) are in the Ginger Family, which would account for the smell of ginger in the root. There are some nice pictures at http://www.azorenflora.de/hedychium_gardneranum.html, which appears to be a web site of flowers from the Azores. The site isn't in English, but that's why there are scientific names. Nice flowers, anyway.