Asked to think back on his coaching encounters with the New Orleans Saints’ Sean Payton, the Tampa Bay Buccaneers’ Lovie Smith did not hesitate to remember how long they have had to test wits against each other in the NFL.

“Sean and I go back ... (to) when he was a coordinator with the Giants and I was (a coordinator) with the Rams,” said Smith, referring to when he ran the defense in St. Louis from 2001-03 and Payton was in charge of the offense in New York from 2000-02. “(Facing Payton means) a lot of formations — innovative thinking — as far as offensive football is concerned.”

But Payton was in less of a mood to reflect on his meetings with Smith as the Saints (1-3) prepared to host Tampa Bay (1-3) at noon Sunday in the Mercedes-Benz Superdome. At least at first — Payton said during a news conference Wednesday that he preferred to focus on eradicating the mistakes that caused his team to lose three of its first four games.

There’s no way to know whether anything other than practicality was at the root of Payton’s choice to not feed any storylines that might portray the Saints’ next game as a chess match between him and Smith. But this much is true: Smith has had the better day on most occasions his teams have clashed with Payton’s.

And that’s something the Saints must change as they seek their second victory of a season that has seen them lose in Atlanta (2-2), Cleveland (1-2) and Dallas (3-1).

They’ll try to do that three days after safety Jairus Byrd — whom the Saints acquired in March on a six-year, $54 million contract guaranteeing him $26 million — tore his lateral meniscus in practice, sending him to season-ending injured reserve.

Payton and Smith split the two times they met as opposing coordinators, with the Giants losing at the Rams 15-14 in 2001 and then beating them 26-21 in St. Louis the following year.

Then, the first three times they met as head coaches, Smith’s Bears ended the season for Payton’s Saints.

Smith and the Bears denied Payton and the Saints what would have been their first trip to the Super Bowl with a 39-14 blowout in Chicago in the 2006 NFC Championship Game. The Bears took the ball away four times from the Saints, who were in their first year under Payton and managed no takeaways.

The Bears and Saints each had three turnovers when they met in the ensuing regular season at Soldier Field, but New Orleans surrendered a 64-yard punt return touchdown en route to a 33-25 setback that eliminated them from playoff contention.

Payton’s Saints won the turnover battle by one when they played Smith’s Bears in Chicago in 2008, but they gave up a 93-yard TD on the opening kickoff, lost 27-24 in overtime and were again eliminated from playoff contention.

Payton didn’t pay Smith back until 2011, when the Saints had six sacks as New Orleans — a little more than a calendar year removed from winning Super Bowl XLIV — pummeled the visitors from Chicago 30-13.

Any number of circumstances have changed since that contest. Smith is part of a different organization, one he joined in January after being dismissed from Chicago and sitting out 2013. The Saints’ roster has transformed considerably, and they have switched defensive coordinators twice.

Yet neither the Saints nor Smith intend to let their experience against each other be in vain.

Payton on Thursday couldn’t help but notice that the Bucs — with losses to Carolina (2-2), St. Louis (1-2) and Atlanta as well as a win over Pittsburgh (2-2) — were in a three-way tie for third in the NFL in forced fumbles with six.

Two of those fumbles have been the work of 2013 first-team All-Pro linebacker Lavonte David, who has recovered one of those loose balls.

Having committed seven turnovers this season and having been defeated by Smith in games where they had too many of those, the Saints realize ball security is even more of a priority than it typically is.

“More than anything else with Lovie, you can see their speed to the ball,” Payton said. “The speed to the ball (has) caused fumbles ... (and) that is something that seems to follow Lovie wherever he goes.”

In his ninth year with the Saints, tackle Zach Strief added that some defenses have a different set for every offensive set out there. But Smith is more rigid with his schemes “because he wants his guys to know exactly what they’re doing, to let them be comfortable in that and to let them play fast,” Strief explained. “So you (get) guys that are decisive, can play hard and ... (aren’t) out of their lines, out of their coverages ... are going to know exactly what they’re supposed to be doing.”

For Smith’s part, it doesn’t escape him that Payton’s Saints offenses have finished atop the NFL four times and have never been worse than sixth, largely because their passing attack has perennially been razor sharp.

Nonetheless, with the exception of the NFC Championship Game in ’06, Payton has dialed up rushing games that have gained 105, 118 and 149 yards for the Saints against Smith.

This year, the Saints are averaging 131.2 rushing yards per game, which ranks 10th in the NFL. It’s that aspect of the Saints that Smith highlighted as a key to success for the Bucs, he said.

“We know them well,” Smith said. “Of course, they know us well.”