Well with regards to your question about what he thinks you have, I think it's pretty unlikely he thinks you have the hand you have. When you check back the turn, your most likely holdings for me are, in no particular order:
1) Two spades
2) Complete air
3) Made house with 44/33
Given that that's the case, he shouldn't really be check/raising for value with 67, so your biggest questions involve what you think he's likely to play pre-flop, and what he's likely to check/call the flop with.
Is he the kind to flat from the SB with

? Would he then flat the flop with any set/2 pair? One hand that makes some sort of sense is

, but by not betting the turn you make it pretty hard to conclusively put him on anything. In your spot I'm just going to ask the question is this player capable of making this move in this spot? If so then I'm going to try and bluff catch.