rhythm dividends

Had a thought on the bus.  If you’re beating time on, say, 1 and 3 with one hand, and 2 and 4 with the other, what happens if you insert a beat at the point where your hands cross in mid air.  You’d get 8th notes out, but how would it work with less regular tempos, and sparser starting points?  And is there a name for conjuring rhythms like that?  It seem like there should be.