I prefer them up over the visor, I don't agree with the stretching comments as you can simply loosen the buckle a notch. One of my relic lids came with a liner and chinstrap that were heavily shrunken and warped. The chinstrap looked like a shrivelled flatworm so I soaked it in leather conditioner until it was pliable and then installed it over the lip where it of course dried out again and assumed its new position where it will remain until it turns to dust.
Having said that. they look pretty good with the strap hanging too. Maybe i'll mix and match the two methods with further lids.