Help with wiring of old lathe

I picked up an old Sheldon EXL 56P 10" lathe the other day. Previous owner put a new Dayton 110/250 motor on it (that's whole nother story). Anyway, he had it hooked up in his garage so I could run it. He had this plug on it (250, 3 prong, twist lock) but the lathe was apparently wired for 110 (according to the add. I didn't get to speak with the owner directly).

The lathe has a fwd./rev. drum switch and ran fine in both fwd. and rev. in his garage when I checked it. Drum switch: I got it home and wanted to stay with 110 as I have lots of 20amp 110 circuits in my shop. So I installed a 110/250 four prong twist lock receptacle in one of the my two gang 110 duplex boxes on the wall where the lathe will go: Removed the three prong 250 plug he had on the lathe and replaced it with a four prong plug that matched the new outlet: like this: Plugged the lathe in and turned the drum switch to forward. The lathe (chuck) ran in the counter clockwise direction. Switched the drum switch to reverse and the lathe again ran in the counter clockwise direction.... huh.

What's up here? Does both the black and white at the receptacle need to be hot for the fwd./rev. to work properly in this type of set-up? My 20 amp 110 outlets are all wired with the typical hot/neutral lash up.

If I really had to, I could run some flexible metal conduit along the wall from a junction box that contains a hot 220 circuit and run the lathe on 220 I suppose, but I was just trying to take the path of least resistance.

Any help would be appreciated.