Hi livergrrl,
I have talked with them and they are fine with any “suitable” foundation which means a foundation which has even and firm support across the entire surface which would include a solid surface, a wood slatted foundation, a wire grid (like they sell on their site) or even a firm innerspring foundation that was evenly supportive and was in good condition with no sags or soft spots (although this could change the feel of the mattress from it’s basic design for some people). They also suggest more closely spaced slats for their mattresses with a latex core. In other words they basically follow similar recommendations to the foundation thread here.
I wouldn’t have any worries about them shifting or slipping on an adjustable bed no. There is a retainer bracket on the end (or sometimes on the corners) of adjustable beds that keeps the mattress from sliding when it’s raised and lowered. The only other thing I would be aware of is that of course there will be a gap in the middle that you will notice between the mattresses if you use the mattresses together and use the middle area of the mattress and that you will need a split king set of sheets if you plan to use common bedding over both mattresses. These have two twin XL fitted sheets and then a king size top sheet. If you are getting two twin instead of twin XL then to do this you would need two twin fitted sheets and then a king would work for the top sheet. Separate bedding would be the same as any twin size mattress.
Phoenix