So far I've found only these short tournament rules:
Two-Headed Ogre Formats: Specific Rules
Each team has two players, designated A and B. These designations are chosen at the start of each tournament and never change during a tournament. Player A is the team captain and makes the final decision in any team disputes. Both players may use the same hero.
Players on the same team sit next to each other. Team members may talk, compare hands, and share information freely with each other. Player A sits to the right of Player B.
Play starts with Player B of team 1 going first and continues clockwise, with each player taking individual turns (1B then 2A then 2B then 1A).
Both opposing heroes must be destroyed in order to win. If a single player is eliminated, play continues clockwise, skipping the eliminated player.
A “turn sequence” is a succession of turns in which each remaining player takes exactly one turn, starting with 1B (or if 1B has been eliminated, 2A, or if 2A has also been eliminated, 2B) and ending with 1A (or if 1A has been eliminated, 2B, or if 2B has also been eliminated, 2A).
Two-Headed Ogre tournaments use single-game matches with a 60-minute time limit. When time is called at the end of a match, players finish the current turn and then play one to four additional turns, depending on when time is called and how many players remain. If time is called during the final turn of a turn sequence, one additional turn sequence is played. Otherwise, the current turn sequence is completed. Then, play stops, and the match winner is determined using the following process:
A team wins if they have more heroes in play than the opposing team.
If both teams have the same number of heroes in play, the team whose heroes have the most combined damage loses. If combined damage is tied, play continues. When there is a change in damage totals, the team whose heroes have the most combined damage loses. This is a pre-priority check.
Two-Headed Ogre Constructed: Specific Rules
Each player on a team has an independent deck and hero. Both of a team’s heroes must be from the same faction.