Try:
SELECT team,
SUM( Wins ) As Wins,
SUM( Losses ) As Losses
FROM (
SELECT homeTeam As team,
SUM( case when homeScore > awayScore then 1 else 0 end ) AS Wins,
SUM( case when homeScore < awayScore then 1 else 0 end ) AS Losses
FROM games
GROUP BY homeTeam
UNION ALL
SELECT awayTeam,
SUM( case when homeScore < awayScore then 1 else 0 end ) AS Wins,
SUM( case when homeScore > awayScore then 1 else 0 end ) AS Losses
FROM games
GROUP BY awayTeam
) q
GROUP BY team
demo: http://sqlfiddle.com/#!15/3db10/11
One remark:
you are using <
and >
operators in queries - this doesn't include cases when awayscore=homescore