NEW YORK — The struggling Chicago White Sox, whose fall out of contention culminated with moving veteran players at the trade deadline, were plagued by a culture that had “no rules” in which a rookie regularly fell asleep in the bullpen, former White Sox relief pitcher Keynan Middleton said Sunday.

Middleton, who was dealt to the New York Yankees just before Tuesday’s trade deadline, joined the White Sox this season after signing a minor league contract and was one of their better relievers prior to the deal. Despite hopes to contend this season, the White Sox are 45-68 under first-year manager Pedro Grifol, whom the team brought in after Tony La Russa’s resignation.

“We came in with no rules,” Middleton said. “I don’t know how you police the culture if there are no rules or guidelines to follow because everyone is doing their own thing. Like, how do you say anything about it because there are no rules?

“You have rookies sleeping in the bullpen during the game. You have guys missing meetings. You have guys missing PFPs (pitcher fielding practices), and there are no consequences for any of this stuff.”

Multiple sources, who corroborated Middleton’s account to ESPN, said a pitcher was seen napping during games as well as skipping fielding practice.

When contacted Sunday night about Middleton’s remarks, the White Sox said they had no comment.

Middleton was traded to the Yankees on Tuesday for right-hander Juan Carela, one of five deals that included veteran starters Lucas Giolito and Lance Lynn, veteran relievers Kendall Graveman, Joe Kelly and Reynaldo Lopez, and third baseman Jake Burger. He spoke after pitching two innings for his new team Sunday, praising the culture set by the Yankees.

“The second I found out I was traded, I shaved my face,” Middleton said. “I was ready to play by their rules because all I want to do is win games. … You know how to act [here]. You know not to be late and you know there are consequences if you are late.”

Middleton said the problems at the White Sox predated his arrival this season — as well as Grifol’s. The team finished 81-81 last year under La Russa, the Hall of Fame manager, before he left the job because of health concerns.

“When I got to spring training, I heard a lot of the same stuff was happening last year,” Middleton said. “It’s happening again this year, so not sure how I could change it. They don’t tell you not to miss PFPs. They don’t tell you not to miss meetings, and if it happens, it’s just, ‘OK.'”

The White Sox struggled in April while dealing with injuries and a tough schedule. In the midst of a 10-game losing streak that month, the team held a meeting in Toronto. Only pitchers spoke up, sources in the room told ESPN.

“I wouldn’t say anything bad about the pitching staff,” Middleton said. “We went about our work the right way. I think the rest of the team struggled to do the right thing.”

Middleton cited players leaving for the World Baseball Classic — particularly Lynn and Graveman — during Grifol’s first spring training as one reason the team’s culture lagged.

“If you’re trying to create culture, you need your big dogs,” Middleton said. “The guys who played in the WBC were our big dogs, and those are the guys I feel like can police the things that are happening.

“There was no jelling of the team. We’re supposed to find our identity in spring training so we can roll out for the season. If you don’t find your identity, you’re scuffling from Day 1.”

Asked where the void exists with the White Sox, Middleton said: “Leadership in general. They say s— rolls downhill. I feel like some guys don’t want to speak up when they should have. It’s hard to police people when there are no rules. If guys are doing things that you think are wrong, who is it wrong to? You or them? It’s anyone’s judgment at that point.”

On Monday, Middleton returns to Chicago for a three-game series against his former team, happy to be in a new setting.

“You hear about it before you come over here,” Middleton said. “Did I want to shave my beard off? Hell no. I had to. I wanted to be a part of [the Yankees]. There was no question.”