[Trix]

When he checks behind on the JT6 twotone board, then it´s often because it looks like a board that hits alot of defending hands and he doesn´t want to pay alot to get to showdown. Mostly he holds something that doesn´t benefit much be getting a cheap rivercard.
Hands like K5s, Ax, Q7s etc are common here.
I think you will win with a bet right there on the turn between a third and half the time, so I´d bet it as I get 2.25:1 on doing so.
If he calls or raise, then you need to think about what kind of player he is.
My default would probably be check-call the river as he could have picked up a draw and probably dont bet weak pairs/A-high when I check, which means bluffs, like Qx or such become a fair part of his range.

gotta go, later.