The schedule below does balance the male-female partnerships, but at the expense of the imbalance of the opposition pairs. Each player has one opposite sex player who they never oppose, and two other players who they oppose twice. For example, consider M1: the pair (M1 F3) never oppose, while (M1 M4) & (M1 F4) oppose twice. Hope that helps.
[(M2 F4):(M5 F3)] [(M6 F2):(M3 F6)] [(M1 F5):(M4 F1)]
[(M5 F6):(M4 F5)] [(M3 F4):(M1 F2)] [(M2 F3):(M6 F1)]
[(M5 F1):(M6 F6)] [(M3 F2):(M4 F3)] [(M2 F5):(M1 F4)]
[(M2 F6):(M3 F5)] [(M1 F1):(M5 F2)] [(M4 F4):(M6 F3)]
[(M1 F6):(M6 F4)] [(M5 F5):(M3 F3)] [(M4 F2):(M2 F1)]
[(M5 F4):(M3 F1)] [(M4 F6):(M1 F3)] [(M2 F2):(M6 F5)]