This continuation c/r is an interesting move, but you need to ask yourself what is the value that he is getting here?
He might have a great hand, in which case he risks losing value when it checks through, although he might pick up value when you bet and he c/c all the way down, of course in that case, why blow you off your hand with a c/r? Of course, he also might have seen you call a c/r previously and call down, but when he doesn't know the strength of your hand, a bet/bet/bet line is likely going to get the most value most of the time.
He could be stone cold bluffing, in which case he is making a riskier play to get you to fold. Odds are, on a board like that, you would fold to a regular c-bet, and although a c/r bluff has the benefit of putting more money in the pot, it also has the disadvantage of forcing him to commit more money to get the fold, therefore, he has to be much more certain that you both will fold. It is a riskier play which requires knowing a lot about your Vs for the benefits to outweigh the disadvantages, imo.
He could be semibluffing, as you pointed out, with the A
X
. This is the most interesting situation and I'm sure he would be happy to take it down right then, but unfortunately for him, this is the exact outcome that would lose him the most value. If the club flush comes in, he would get at least some value from you (might stack ya) in the flush over flush scenario.
This board is so dry, imo, that he is targeting too small a range for a value c/r, and is risking too much for a bluff/semibluff c/r. The board would have to be a lot wetter for a c/r as the PFR to have any merit.