  For a multiplicative cohomology theory E, complex orientations are in
bijective correspondence with multiplicative natural transformations to E from
complex bordism cohomology MU. If E is represented by a spectrum with a highly
structured multiplication, we give an iterative process for lifting an
orientation MU -> E to a map respecting this extra structure, based on work of
Arone-Lesh. The space of strictly commutative orientations is the limit of an
inverse tower of spaces parametrizing partial lifts; stage 1 corresponds to
ordinary complex orientations, and lifting from stage (m-1) to stage m is
governed by the existence of a orientation for a family of E-modules over a
fixed base space F_m.
  When E is p-local, we can say more. We find that this tower only changes when
m is a power of p, and if E is E(n)-local the tower is constant after stage
p^n. Moreover, if the coefficient ring E^* is p-torsion free, the ability to
lift from stage 1 to stage p is equivalent to a condition on the associated
formal group law that was shown necessary by Ando.
