Yes, the union (or more specifically the SIG - Schedule Improvement Group) looks at every pairing.

The company can build a pairing that is legal under the contract but the union (ie SIG) can dispute it. This normally happens when the pairing is unrealistic (ie a 35-minute scheduled turn in BOS at 0730) or especially onerous (ie CDG-STN-MEM in one fell swoop).

When the union disputes a pairing it cannot be built into lines. The only way a pilot can voluntarily fly a disputed pairing is to specifically request it (either on a secondary line or via trip-trade or make-up). Ideally all disputed pairings would be flown by guys on reserve, and they might call in fatigued every once in a while.

What happens instead is that greedy clueless bastards see something that works for them (a disputed pairing that fits into their schedule) and then pick them up and fly them. This diminishes any leverage the union might have with that particular pairing, and after a few months of greediness that ugly pairing might start showing up on lines.