hojack, it shouldn't make a difference which side of the 240-volt line your taking power off--the white neutral shouldn't be overloaded IF the house is wired correctly
(I have come across a situation where a licensed but incompetent electrician used a 3-wire-plus-ground approach ("shared neutral") but put both breakers on the same side of the 240-volt line; that is wrong and a fire hazard!)
With respect to phasing, doesn't make any difference whether the transformers are on different circuits; what does make a difference is whether both circuits come off the same side of the 240-volt input.
In one sense phasing does relate to polarity. Polarity on a 60-hertz line changes 120 times per second, and phasing means that both outputs are positive and of equal voltage at the same time. (Dale is correct that here we only have one possible misphasing 0 or 180)
The polarity of the output is dependent on the polarity of the input at any particular instant. If the 120-volt AC line at a particular INSTANT is feeding positive to the left side of each identical transformer primary, the outputs should be in phase--if to opposite sides of each transformer primary, then they're out-of-phase
As to your question about the clipped plug, I expect that the transformer internally does not have any connections to its frame--that the two conductors go only to opposite ends of the primary coil--so it probably shouldn't make a difference. PROBABLY!! But be sure to mark the plug so you always get it right.
