I recently played a tournament where teams of four competed against each other. There were nine players and nine rounds, so each player sat once. The goal was that each player play with and against each other player the same number of times. Since players have three teammates and play eight rounds for a total of 24 partners, the collection $\mathcal{T}$ of teams could be interpreted as a $2 - (9,4,3)$ block design. However, there's an additional constraint:
To determine the matches, we need an involution or matching $\tau: \mathcal{T} \to \mathcal{T}$ so that for $T \in \mathcal{T}$ we have $\tau(T) \cap T =\varnothing$.
We may also need to specify that each player play each other player the same number of times. In the case with 9 players, this occurs automatically since for each pair of players, they play together 3 times and each takes one distinct game off and thus play against each other four times. However, as we increase the number of players relative to team size, we may wind up with the same players sitting multiple times, which would be an issue.
Do these sorts of constraints appear in the literature on block designs? Are there easy ways to construct these sorts of matchings for designs in general?