The House is considering whether to have a single vote on two bills at once: (1) the Senate Health Care Bill and (2) the Reconciliation Bill.
The two bills that the House is now contemplating directly contradict each other. For example, one says that Nebraska gets the “Cornhusker Kickback” and the other says Nebraska does not get the “Cornhusker Kickback.”
I think the constitututional principle being violated here is known as the “nondelegation doctrine.” By saying two contradictory things at the same time, the House would be delegating its power to the Senate and the White House, allowing the latter to pick which meaning they like best.
Congress could use the same approach to allow a line-item veto, by passing a thousand budgets instead of one, and letting the President pick whichever one he likes best. And that would also violate the nondelegation doctrine.
UPDATE (3/14/2010): Just to clarify, I'm skeptical that the judiciary would get involved based on this nondelegation argument, even though I think it's a correct argument. On the other hand, I do believe that the judiciary might very well get involved based on the following constitutional provision:
Every Order, Resolution, or Vote to which the Concurrence of the Senate and House of Representatives may be necessary (except on a question of Adjournment) shall be presented to the President of the United States; and before the Same shall take Effect, shall be approved by him…
So, if the House votes on two bills at once, the President must approve that vote “before the same shall take effect,” because the vote at least partially requires the Senate's concurrence. The President cannot pick and choose which part of the House vote to approve, and the House vote cannot have any effect unless the President approves it.
The framers saw all this coming. You can read about it here in the Heritage Guide to the Constitution.
UPDATE #2 (3/14/2010): Former US Circuit Judge Michael McConnell has weighed in with an opinion piece in the Wall Street Journal arguing that the Slaughter Solution is unconstitutional.
UPDATE #3 (3/16/2010): Several more constitutional scholars are voicing doubts about the Slaughter Solution. I have yet to hear about any precedent in the entire history of the United States for the Slaughter solution: the House votes for two bills in a single vote, only one of which is approved by the Senate, and the President signs that one bill into law.