"There is only one way the Wingman will work. You can't turn it (or the Batwing) the other way."
The "levelling" bracket that fits over the two masts ( "lift tubes") has holes in it at each end with two pins to hold it on. I got the antenna head on the other way around by using the bracket the other way around.
However-- that bracket came with the Jack so you can put the Jack on a Winegard mast, so I can't confirm the OEM batwing's bracket does too. I think it does. Drawing in manual not clear but shows perhaps three holes, so it looks like it would do it. (EDIT--needs confirmation--see edit below.)
The mast still went up and down, but when down, the antenna head was pointing the other way.
EDIT--OK I dug out the Jack manual, and it's bracket has four holes so "antenna may be mounted in either direction in order to clear obstacles such as AC units or roof vents". I am using the Jack's bracket.