Actually, upon further review, I don't think it's a hydrostatic. It's the original H1011 SA.
So, the shift lever has a "Reverse-Neutral-1-2-3-4-5" speed selector, is that right?
When you move it to "Reverse" the mower moves forward, but when in "1-2-3-4-5" it goes in reverse, am I understanding that right?
Did the seller give you any clues or hints about this problem before you bought it? Any ideas about what might have been done to the unit before you bought it? Service records, etc.?
I don't think it is mechanically possible for the shifter to be connected backwards. If the transmission has been apart, could be something internal. The transmission is driven by a short belt off the engine crankshaft; I take a wild guess and wonder if the belt is somehow in a figure-8 position, which would reverse the operation of the transmission, but that is a longshot for sure.
Some photos of the underside showing the belt between the transmission and engine crankshaft would help...