You could use the 4 directions as a proxy for political axis (liberal - conservative but with 4 poles instead of just 2 -- its space, so clearly more complex). Each senator has a particular direction or 2 that they are naturally aligned with. Perhaps if they are on the wrong edge or edges of a debate at the end they get lopped off between rounds (they are defeated in the next election / recalled by their planet...).
You could have 2 or 3 rounds to each debate to simulate the debate moving from sub-committee to committee to full senate vote. Each round ends after a certain number of cards played, then edge senators are checked to see if they get recalled. Then you continue to the next round.
To stay true to that theme final scoring could simply be that each senator = 1 point, so most senators aligned with you win (before or after edge senators are checked for recall after last round?).