My GSD Echo also sleeps on her back. Legs splayed. :0
We are looking after a Jack russel / chihuahua mix called Max for a couple of weeks and Daughtry
our (2yr old Husky female) got down on the ground and shuffled forward towards Max.
At first he was VERY wary / scared of all the dogs.
Daughtry did this to put him at ease, "look I'm not a threat, I'm not going to dominate you I just want to be friends."
Soon she was sniffing and nuzzling him. It's non threatening way some dogs use to approach others.
also, It's not necessarily a submissive gesture, just non threatening.