The Texas Republican senator Ted Cruz called a new highway project “a great bipartisan victory” that will bring “jobs to Texas and millions of dollars to the state”.

The White House responded: “Senator Cruz voted against this.”

A spokesperson for Cruz said the senator “made it possible” – but did not contest that Cruz voted against the $1.5tn spending package which contained the highway project.

With Ben Ray Luján, a New Mexico Democrat, Cruz co-sponsored an amendment adding the Ports to Plains highway project to the spending bill earlier this year.

Earlier this month, Cruz tweeted: “The Ports to Plains highway will run from Laredo all the way up to North Dakota and into Canada. This project will bring jobs to Texas and millions of dollars to the state. A great bipartisan victory!”

After the White House hit back, an anchor for KAMC News, a Texas TV station whose interview Cruz used in his tweet, shared footage of a follow-up question.

Ryan Chandler asked: “Why shepherd this federal highway designation through the process and then end up voting against the actual legislation that put it into law?”

Cruz said: “That happens frequently in the United States Senate, where you end up working to get agreement and to pass a particular piece of legislation, but then it gets rolled into a giant bill that has a whole bunch of good things and bad things.

“There have been dozens of different pieces of legislation that I wrote, that I got support for, that I got passed into law, but the ultimate vehicle they got stuck into had other elements that were bad and wasteful and didn’t make sense, so I’d vote against the giant mess of a bill but at the same time, enact the legislative victory that’s focused on jobs in the state of Texas.”

The other Texas US senator, John Cornyn, voted for the spending bill. He said then: “The Senate has the responsibility not just to keep the lights on but also to make critical investments in our country.

“… Despite its flaws, despite the crazy process by which we find ourselves here voting on this $1.5tn appropriation bill, notwithstanding all the reasons I could cite why maybe I should vote against it, I think there’s enough good in this bill to support it.”

On Wednesday, Cruz’s spokesperson stuck to Republican midterms messaging when he called the appropriation bill “a Democrat spending spree that contributed to an economic recession for American families”.

But Jaime Harrison, chair of the Democratic National Committee, was happy to offer Cruz “a civics lesson”.

“When you vote ‘NO’ on legislation it means … wait for it … you DO NOT SUPPORT the bill, therefore you should not take credit for something you didn’t vote for and support.”