There are two different flags at issue here–the upside-down American flag, and the "Appeal to Heaven"–the former was flown at their main residence, the second at their vacation home. Both have long histories of being used all over the political spectrum (both right and left), to mean lots of different things.

The article you are citing is relying on the claims of an angry former neighbour. Why assume that the Alito's are lying or mistaken, when it seems just as possible that their ex-neighbour could be.

In any event, even if we assume the ex-neighbour's claims are 100% true, they still don't prove (1) that Martha Alito flew the flag with intention to express a highly specific political message, (2) even supposing that was her intention, it still doesn't prove Samuel Alito was supporting her expressive act, as opposed to simply allowing his wife to do what she wants to do.