Normal 3-ways are wired like this
Your first part suggests a 3-way circuit where one of the switches is miswired.
There is such a thing as 3+ switches on a 4-way circuit. Thise switches are added to the middle, and take 2 yellows on one side and 2 yellows on the other side. They either send them straight through, or swap them.
I use yellow for travelers because they are a matched pair that are interchangeable. Practical 3-way wiring with cable forces you to use 2 of the 3 available wire colors for travelers, and it can be different on every wire run. As such, I recommend removing all uncertainty by marking them with yellow tape. It makes complicated things dog simple.