Reading the text of that along with the President's adjournment power would lead me to believe that a disagreement would not require a proactive action of both Houses. Let's say the House passed a resolution to adjourn for two weeks. The Senate fails to take up the resolution. At that point, the House has already adjourned itself, while the Senate is still conducting business. After three days have passed, I think there would be three possible outcomes. The least controversial action would be that the Senate could take up the resolution and agree to adjourn. The other two options would be that the House agrees to return or that the House refuses to return and the President declares that a disagreement has been established and thus adjourns Congress.
Based on the text of the Constitution, if one House of Congress passed an adjournment resolution and the other failed to act on it, I would think the President would be fully empowered to exercise his authority after three days.
Fourth option: the President could use the other power in the excerpt you quoted above and force the House back into session, right?
