Algorithm to organize a tournament where the team componentes change each round
Sorry, but there is no solution. (Assuming that all games are 2 vs 2.)

How many matches does each player play?


They play with each other player exactly once, one such player per match, for a total of $n-1$ matches.
They play against each other player exactly once, two such players per match, for a total of $\frac{n-1}{2}$ matches.


This implies $n-1=\frac{n-1}{2}$ and thus $n=1$. But you said $n$ is even. And besides, a single-player tournament would not be interesting.

[EDIT]

The question was now changed to omit the requirement that every other player is a partner at least once. However, it is still not possible.

How many rounds should there be? For every player there are $n-1$ potential opponents. In every round there are $2$ opponents without repetition. Hence the number of rounds is $\frac{n-1}2$. This is not an integer number, because $n$ is even.