Golly, it sounds like the way they wired it, they are switching neutral.
Since a 3-way is involved, and there aren't any /3 cables coming through here, then either a) the switch group supplies power, or b) the switch group is a spur switch loop.
a) (switch group being power source) doesn't sound right because why have 2 more cables going off other directions? You don't have 2 other lights under this same control. So that leaves b).
What makes the most sense is that the 3 cables are the following:
- supply from panel
- onward power going to other outlets
- switch loop.
The first 2 cables' blacks are always-hot (note the fan is connected to this).
The first 2 cables' whites are actual neutral (note the fan is NOT connected to this).
The third cable is the switch loop, black and white are connected or not.
So power goes from always-hot -> to the fan -> to the switch group on black -> back to neutral on white.