Both sides have points but we’ll try to sort some things out. Woodward’s basic substantive charges about the sequester are that it was the administration’s idea to begin with, and that the White House has “moved the goal posts” by now insisting on new tax revenue as well as spending cuts to reach further deficit-reduction goals.
The former is true. Administration officials did come up with the sequester idea as a way to try to force Congress to agree to those deficit-reduction figures. The White House pretty much acknowledges this but adds that it is kind of irrelevant, because it was House Republicans who were holding the debt-ceiling increase hostage at the time. If Speaker John Boehner et al had not been doing that there would have been no need for sequestration, Q.E.D. Plus, it wasn’t supposed to ever go into effect.
The latter Woodward charge is more open to debate. Speaker Boehner may have thought that the “grand bargain” deficit-reduction package he nearly struck with the White House back in 2011 was all about budget cuts. But the administration pretty clearly thought more tax revenue would be included as well.