No it doesn't, when one cylinder is on its firing stroke, the other is just finishing exhausting and beginning the inlet stroke; there is one firing pulse per crank rotation, if it worked the way you suggested, there would only be a firing pulse per two crank rotations (as a single cylinder four stroke does).
The way I described was the way it was described to me, but like I say I probably have the wrong end of the stick,
I just had a look at the Video again of the TA Cutaway, and like you say from the Valve/Cam positions it seems to be on alternating cycles.
Still I am confused.