I have a little doubt if this structure is actually correct or not. I think the more standard alternative is to say:
Did you go to school or stay at home?
However, they do not feel quite the same to me. I cannot exactly pinpoint the difference, but to me, it feels like the second one suggests a more inconsequential choice between going to school or staying at home. On the contrary, the first one sounds like the default option is to go to school but it is also possible for people to stay home for reasons such as being sick.
Can someone give me some perspective on the grammatical correctness of these two structures and any difference in their meanings?