While I don't have any direct experience with LaCrosse, here are a few thoughts gleaned from designing my own and working with other equipment.
The Wind Vane (the arrow thingy that shows direction) is likely an analog voltage, from 0 to whatever the supply voltage is.
The Anemometer (the cups or turbine) is likely a voltage pulse proportional to the wind speed, probably one pulse per revolution of the cups or turbine blade.
If you have a 4 wire connection to the wind unit, one wire is probably ground, and one is the supply voltage (that one would be constant). On a Davis VP2, it's about 5 volts.
One of the other wires would be the wind vane and would vary between zero and the supply voltage depending on the position of the vane. If I were designing it, North would be the transition between 0 and full voltage, and half of the supply voltage would be due South. east would be 1/4 voltage.
And the last wire would be the 0 to full-voltage pulse for the anemometer. That's assuming the wind speed sensor is a rotating magnet and a reed switch.
There are more elegant and expensive ways to do it, but for cheap (LaCrosse) I'd bet that this is how it works.
Be clear what you are measuring. "Vane" = wind direction. "Anemometer" (cups or turbine/propeller) = speed.
Turned slow enough (ie: by hand) you should be able to see the anemometer pulses on a DMM or VOM.