According to my somewhat-ancient (1978)unabridged World Book Dictionary (L-Z)...

Rigadoon,n.
1 a lively dance for one couple.
2 the quick, duple rhythm for this dance.
3 a piece of music in such time.
French rigodon, rigaudon; origin uncertain; probably from Rigaud, a Marseilles dancing master, who invented it

I tend to concur with Dulcimer, since the beggarman COULD well be dancing with himself (def. 1) sans partner while singing as in definition 2. To top it off, the way I have heard (and learned) this song it could well qualify itself under def. 3! =)