ybc, yes you can adjust your property tax down in case of depreciation. That’s proposition 8 — temporary reduction of assessed value for property tax purposes.
However, when prices go up again, your house will automatically revert back to the original basis + 2% annual compound. Therefore, for property taxes purposes, it’s always better to buy low and hold-on forever. When you retire (after 55, I think), you can transfer the original basis to another house of equal or lesser value.