With the new Congress convening, incoming Senate majority leader Mitch McConnell (R-Ky.) has been trying to set a strikingly new GOP tone, indicating that he wants to pass legislation in cooperation with Democrats instead of merely running against them.

“I want the American people to be comfortable with the fact that the Republican House and Senate is a responsible, right-of-center, governing majority,” McConnell said. “There would be nothing frightening about adding a Republican president to that governing majority.”

If acted upon, McConnell’s words indicate a much more positive direction for divided government over the next two years, in contrast with the past four. Let’s hope the party follows the Senate’s new leader.

But two things. First, no one should imagine that a GOP Congress would act the same way without the check of a Democrat in the Oval Office. McConnell himself said as much after describing what sorts of measures could lead to compromise: “Could the country use a lot more? You bet. But there’s no way you can overcome a reluctant president on something really large.” The 2016 election must be about the full Republican agenda — including how much of the Obama legacy voters really want repealed — not just whether voters are sick of the president or whether Republicans haven’t been “frightening” for a while.

Second, assuming the House GOP follows McConnell’s lead, no one should be comfortable with the way Republicans came into their new attitude of responsibility. Neither party should believe that the only time it has reason to behave responsibly is when it controls a branch of government. The system would barely function — like it has barely functioned for the past several years.

A more orderly Congress would be a big improvement, as would progress on trade, transportation and other areas of potential compromise. But Republicans aren’t starting with a clean slate. They begin their rule of Capitol Hill marred by a reckless recent past — in some of the policy they have pushed and, particularly, in the tactics they have used. McConnell and the GOP will have to work really hard over the next two years — in ways that make the right wing a lot more uncomfortable — to demonstrate that Republicans deserve more trust.