Using a third party search engine to sort this list of sentences that sounds right to me, I’m a native English speaker, and many who have used this site have no issues. That is because the common phrase construction is

Neither nor .

So you could say something like

We saw neither the castle nor the cathedral, and we would rather not kill ourselves, “We saw neither the castle nor the cathedral and how do we know it”?!

Or you could say

We didn’t get to see either the castle or the cathedral.

How can you remember that by way of saying that ‘didn’t’ but you already had a negative and so using didn’t with neither is wrong?

Neither/nor is the opposite of either/or or both/and.