I have now got these new sideview labels, after a bit of a delay (and some Bad Gateway errors that might be due to all this tinkering with the innards of Ancestry).
The ones I allocated are now labelled as Maternal and Paternal instead of Mother's and Father's side. With them I get told which of Parent 1 and 2 they are (when Ancestry thinks it knows), and I'm offered this option:
This match is on your Parent 2's side. Do you want to change Parent 2 to Paternal side? This will label all Parent 2 matches Paternal side and Parent 1 matches Maternal side. Update label
I haven't done that yet - I want to know if that's reversible, or if I will still be told which I labelled and which Ancestry's worked out. I can't find any help on this topic yet, so I guess I can leave it for now - I know which parent is which anyway.