We can use a SELECT statement with a GROUP BY school to create a record for each school. The ties, wins, and losses columns are readily calculated with the SUM aggregate function, as you noted. To target a specific round, we can use some clever math (to avoid verbose conditional statements like the one CodeByMoonlight suggested):
If we want to target round R, we note that "round-R" is 0 only when round == R, otherwise it isn't 0. When we take the NOT of "round-R", 0 gets inverted to 1, while everything else gets set to 0. Now, if we multiply !(round-R) by the score of that round, it will give us 0 when the round is not R (as 0*score = 0) and it will give us "score" when the round is R (as 1*score = score). Next, when we take the SUM of this value over the columns, we add score when round=R and 0 otherwise, effectively giving us just the round R score.
Putting that all together gives:
SELECT school AS `School`,
SUM(!(round-1)*score) AS `Round1`,
SUM(!(round-2)*score) AS `Round2`,
SUM(!(round-3)*score) AS `Round3`,
SUM(!(round-4)*score) AS `Round4`,
SUM(!(round-5)*score) AS `Round5`,
SUM(!(round-6)*score) AS `Round6`,
SUM(!(round-7)*score) AS `Round7`,
SUM(!(round-8)*score) AS `Round8`,
SUM(!(round-9)*score) AS `Round9`,
SUM(!(round-10)*score) AS `Round10`,
SUM(win) AS `wins`,
SUM(loss) AS `losses`,
SUM(tie) AS `ties`
FROM `RoundScores` GROUP BY `school`
where RoundScores is the table in question.
EDIT:
If we do not want to manually add 10, we can use prepared statements :
# Store all the conditionals in a string:
# I was not able to to have round loop from 1 to 10, so I iterated over
# all distinct values of 'round' present in the table.
SET @s = "";
SELECT `round`, (@s := CONCAT( @s , "SUM(!(round-",round, ")*score) AS `Round",round, "`," )) FROM `RoundScores` GROUP BY `round`;
# Combine the conditionals from before with the rest of the statement needed.
SET @qry = CONCAT("SELECT school AS `School`,",@s,"SUM(win) AS `wins`,SUM(loss) AS `losses` FROM `RoundScores` GROUP BY `school`");
# Prepare and execute the statement.
PREPARE stmt1 FROM @qry;
EXECUTE stmt1;