Mo's question what would you tie it to if not a stick? makes sense if you've never seen a swag. In Oz, a swag is your bedroll and, traditionally, was not bulky the way modern ones are; thy used to be ~6-8" in diameter and 2-3' long when rolled tight.

The most comfortable way to carry a traditional swag was to centre it on one shoulder so its weight was balanced fore and aft. These day, with vehicle transport, its rare to see one less than 2' in diameter and 4' wide when rolled up.

Rigadoo as a stick rather than as a dance? Many's the time I've danced (in play, mind) with a stick as my partner, so I see no conflict arising from dulcimer's scholarly etymology and others' view of it as a stick.

Cheers, Rowan