  Factorization theorems underly our ability to make predictions for many
processes involving the strong interaction. Although typically formulated at
leading power, the study of factorization at subleading power is of interest
both for improving the precision of calculations, as well as for understanding
the all orders structure of QCD. We use the SCET helicity operator formalism to
construct a complete power suppressed basis of hard scattering operators for
$e^+e^-\to$ dijets, $e^- p\to e^-$ jet, and constrained Drell-Yan, including
the first two subleading orders in the amplitude level power expansion. We
analyze the form of the hard, jet, and soft function contributions to the power
suppressed cross section for $e^+e^-\to$ dijet event shapes, and give results
for the lowest order matching to the contributing operators. These results will
be useful for studies of power corrections both in fixed order and resummed
perturbation theory.
