I checked with the manufacturer. First they told me no, each hot had to be
to a different leg of 240v.

I asked him why, since the neutral can handle the 1600w. He then agreed
that for 1600w it was fine.

The problem is that someone might be dumb enough to try to run 5000w 120
through it, and the neutral would be overloaded; but with a 5-20 input, that
should not be an issue.