This should work, using a self-join could result in many duplicated entries
SELECT
t1.battle_id,
t1.winner,
t1.loser
FROM
your_table t1
WHERE
EXISTS (
SELECT
1
FROM
your_table t2
WHERE
( ( t1.winner = t2.winner
AND t1.loser = t2.loser )
OR ( t1.loser = t2.winner
AND t1.winner = t2.loser ) )
AND t1.battle_id <> t2.battle_id
)