Mirage 6-12: Value Bet on the River?

[Dynasty]

This is a hand which has stayed in my mind despite it being played many months ago. David Ottosen's recent posts about river decisions prompted me to finally post it.

Everybody folds to and MP (cutoff-1) who opens with a raise. Cutoff folds. I'm on the button with 9c9d and 3-bet. Small blind folds. Big blind cold-calls two bets. MP calls.

The flop is: Th,6s,3d

Big blind checks. MP checks. I bet. Both players call.

The turn is: Th,6s,3d,2s

Big blind checks. MP checks. I bet. Both players call.

The river is: Th,6s,3d,2s,Ts

It gets checked to me. Should I bet? Of course, comments on all other streets welcome too.