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Right now (to my knowledge) there is no way to move a drawing locked into the horizontal or vertical plane.
Adding two handles at the center square (which currently serves no function other than a center point) would allow much ease in dragging an image you want locked to only vertical or horizontal movement, vs trying to freehand it by approximation.
Also is there a button I can press currently to do this? In photoshop it's holding shift during movement to do this behavior. Perhaps adding the shift function would also help? Unless TVP has a different button it uses.

On Win / Linux it's [CTRL + 2 / 4 / 8 or 6]
On Mac, it's [CMD + 2 / 4 / 8 or 6]

+1 for dragging with SHIFT to keep it strictly horizontal/vertical

(just like Shape transform: Pan)

I'd really like to have more options in the Transform tool, since I often do some quick and dirty animation just with it. Mostly I miss non-proportional scaling, but being able to distort all 4 points would be nice.

Restricting translation to one axis should be standard in all transform and warp tools, just by holding shift.