There's no control logic relay, no alternating relay, no capacitor. It starts and stops with a single pole switch. I watched it come to a complete stop each time I turned the switch off, and each time when I turned it back on, it went in the opposite direction. We're going to replace the motor, but we both would really like to know what's happening here.
I'm still not clear on whether this is SUPPOSED to do this, or you consider it an anomaly. It sounds more like you think it's an anomaly. But without knowing more about the motor, it's hard to imagine how that would happen. Reversing a
standard single phase motor requires swapping wires in the motor connection box. If nothing is making that swap, the only other possible explanation is the one put forth earlier, that the motor direction is being pre-determined by the air flow (it's called "windmilling"). In a single phase cap-start, split phase or shaded pole motor, it is definitely possible for a windmilling motor to continue running in the wrong direction if started that way.
But... if it is one of the new EC (Electronically Commutated) motors being used a lot in fans now for energy efficiency, it is possible that there is some control logic inside of the fan that is making it reverse alternately. No idea why you would want that on a fan, but that may be where the glitch lies. These are essentially brush-less DC, synchronous or even switched reluctance motors, but the issue is that they all will have an electronic controller built-in to the motor housing. Something may be wrong inside that controller that is causing this to happen.