Sometimes we hear about athletes trying to lose:

Norio Sasaki, the coach of the Japanese women's soccer team, fielded a weaker squad against South Africa and played for a tie as the country tried to sidestep a victory that would have meant a long trip for a match with tougher opponents.
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The badminton teams also may have been trying to lose so they could avoid more difficult opponents in the quarterfinals, when their competition also moves to a knockout stage.

You'll notice, however, that they're not really trying their hardest to lose. When you have your stronger players stay on the bench but still keep going for goals you're balancing your desire to lose against your need to look to the judges like you're still trying to win. But what if we relax the second requirement to only apply to intentional resignations, forfeitures, and fouls? What if we had a game where both sides were trying their absolute hardest to make the other side win?

Basketball

Playing basketball to lose is basically basketball with the baskets reversed and two major changes. The first is that instead of the team that just scored giving up the ball they get to keep it, changing the game from "losers take" to "winners take". There's a theoretical argument that this does not affect who wins the game, except that instead of simple "winners take" this is "winners take, starting under the basket they're trying to score on". This second change means we should get positive feedback. Most games have either neutral feedback or negative feedback: if they're going to give an advantage they give it to the side that just lost (or that is losing overall). In backwards basketball we should see big swings in points.

Hockey

Hockey games start with a face-off and use another one after every goal. The referee drops the puck, and two players compete to gain control of it. While winning face-offs isn't normally a huge advantage, in playing-to-lose hockey it's basically the only thing that matters: once you win the face-off you pass the puck back to team members who score an own goal.

Ski Jumping

In ski jumping you earn points for accuracy and style, so to minimize points you want to jump as short a distance as possible with maximally poor form. Instead of the current highly aerodynamic position used by jumpers you'd want to stand up catching as much wind as possible, but in an awkward enough position that the judges didn't think you had good form. Possibly you just fall over immediately, and slide down the track in an undignified way. In the air you want to look confused and ungainly, and you definitely don't want to land on your feet.

Football

I really don't know football rules well, but I think you end up turning the game into a competition to get possession after a kickoff. As soon as you get possession you make an incomplete pass, and then when play starts again you can run back to your end-zone to get a safety and score the other team two points. While the team being kicked to has the advantage on a kickoff, their success rate should be far enough from 100% (and 0%) to get an interesting game: that the ball has to go at least 10 yards favors the receiving team, but that the kicking team knows exactly where the ball is aimed favors the kicking team.

Boxing

The game turns into a competition to see how fast you can knock yourself out.

Soccer

The team starting with the ball must kick it forward across the center line, and then can't touch it again until another player on their team has touched it. Players from the opposing team have to stay outside the center circle until that first touch, and everyone has to start on their half of the field. Scoring an own goal is then a matter of a very short pass forward across the center line to a player on the same team, followed by a series of quick passes back to the goal. It's not practical for the other team to interfere here, but you would want to be very careful to keep the ball from going out of bounds. After an own goal you again start with the ball, so this would be repeated for an entire half. To begin the second half the other team takes the kick-off, and again they never lose control of the ball. The game turns into a competition to see which team can score the most own goals in 45 minutes without interference from the other team.

(The closest thing to this historically might be the 2002 game between Adema and SOE where SOE protested a decision in a previous game by scoring 149 own goals.)

Bobsled

Teams with specially crafted unaerodynamic sleds would steer side to side trying to lose speed. This is a lot like a slow bicycle race, but downhill on a slippery surface.

Poker

This is going to depend on the variant, but let's say Texas hold'em. Any time you have the option to raise you want to check, because otherwise your opponents will all fold leaving you with the pot. But there are two players who have to put in money regardless: the small and big blinds. What matters here is the big blind, because the small blind can just fold without matching the big blind. If your cards are bad enough you should match the big blind's bet and hope to lose $1, but if you're wrong you gain $3. The game moves through money very slowly and lacks the player interaction of a standard game, but there's still strategy in deciding whether your hand is bad enough to bet and how to interpret other players' deciding to bet.

Quidditch

A game of quidditch runs until one of the two seekers catches the snitch, which earns them 150 points and ends the game. It's not possible to force the opposing team's seeker to catch the snitch, so to end the game you have to be willing to accept 150 points. You can earn 10 points for your opponents by putting the quaffle through one of their goal hoops. Playing to lose you must first gain a lead of 15 goals for the other side and then catch the snitch. It's not clear to me whether the rule prohibiting stooging covers knocking aside a keeper who is defending their opponents' goal. If it does not then accumulating a 15 goal lead should be relatively easy, especially given that after every point the scoring team's keeper throws the ball in. This gives the a positive feedback mechanism, which means a 15 goal lead isn't that hard. It also keeps a lucky play by a seeker from leading to a surprise upset.

Others?

What other games would be interesting in reverse? Baseball, cricket, badminton, most races, and darts don't end. Bowling ties 0-0. [1] Others? What have I gotten wrong above?

(Thanks to Ben Orlin for thinking a lot of these through with me.)

[1] Bowling with bumpers might be interesting, but good players might be able to bowl 30 times in a row without hitting any pins.